The Epistle of the Number by Ibn al-Aḥdab 9781463204174, 2015019326, 1463204175

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The Epistle of the Number by Ibn al-Aḥdab

Perspectives on Society and Culture

2 Series Editor Amir Ashur

This series explores societal issues in the history of the Near and Middle East, from antiquity to the medieval period. Volumes will include monographs and collections of peer-reviewed essays on aspects of community, family life, legal traditions, and economic affairs. Gorgias particularly welcomes proposals investigating aspects of daily life and sectors of society less visible than others in the historical record.

The Epistle of the Number by Ibn al-Aḥdab

The transmission of Arabic mathematics to Hebrew circles in medieval Sicily

Ilana Wartenberg

  2015

Gorgias Press LLC, 954 River Road, Piscataway, NJ, 08854, USA www.gorgiaspress.com Copyright © 2015 by Gorgias Press LLC

All rights reserved under International and Pan-American Copyright Conventions. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise without the prior written permission of Gorgias Press LLC.

2015

‫ܛ‬



ISBN 978-1-4632-0417-4

Library of Congress Cataloging-in-Publication Data Wartenberg, Ilana, 1970The epistle of the number, by Ibn al-Ahdab / by Ilana Wartenberg. pages cm. -- (Perspectives on society and culture) ISBN 978-1-4632-0417-4 1. Isaac ben Solomon Ibn al-Ahdab, 1340 to 1350-1429 to 1433. 2. Mathematics, Medieval. 3. Mathematics, Jewish. 4. Numeration, Arabic. 5. Mathematics--Italy--History--To 1500. 6. Mathematics--Italy--Sicily--History. 7. Jews--Italy--Sicily--History. 8. Intercultural communication. I. Title. QA23.W37 2015 510--dc23 2015019326 Printed in the United States of America

TABLE OF CONTENTS Table of Contents ................................................................................................................... v Acknowledgments.................................................................................................................. ix Foreword ................................................................................................................................. xi Hebrew Abbreviations ......................................................................................................... xv Hebrew and Arabic Transliteration.................................................................................. xvii Chapter 1: Ibn Al-Adab ....................................................................................................... 1 Introduction .................................................................................................................... 1 Isaac’s life and times ...................................................................................................... 1 The historical circumstances ........................................................................................ 3 Castile ......................................................................................................................... 3 Aragon ....................................................................................................................... 4 The Jewish community in Sicily............................................................................. 5 People in direct contact with Isaac.............................................................................. 7 The RiBaSH (Valencia 1326–Algiers 1408) ......................................................... 7 The RoSH’s family: Judah ben Asher the Second .............................................. 7 Samuel Ibn ara (Valencia, second half of the 14th century) ........................... 8  n Danon .................................................................................................. 8 Isaac’s family ............................................................................................................. 9 Where was Isaac in 1391? ........................................................................................... 10 Isaac’s reading sources .......................................................................................... 11 Isaac’s works ................................................................................................................. 13 Mathematics ............................................................................................................ 13 Astronomy .............................................................................................................. 14 The Jewish calendar ............................................................................................... 16 Commentaries and exegeses ................................................................................. 17 Poetry ....................................................................................................................... 18 Medicine .................................................................................................................. 19 Isaac’s linguistic skills as manifest in the Epistle .................................................... 20 The echoes of Isaac’s work ........................................................................................ 20 Summary ........................................................................................................................ 22 Chapter 2: The Hebrew and Arabic Epistles .................................................................... 25 The Arabic source ........................................................................................................ 25 Ibn al- 

 .......................................................................................................... 26 Naming the Hebrew Epistle................................................................................. 27 The Hebrew and Arabic texts compared ........................................................... 27 The readership and propagation of the Hebrew and Arabic texts................. 27 Table of contents of The Epistle of the Number .......................................................... 28 v

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THE EPISTLE OF THE NUMBER

Book I: Arithmetical operations on integers, fractions and roots .................. 28 Book II: The rules which enable to obtain the unknown from the given known ................................................................................................... 30 Chapter 3: A Mathematical Commentary of The Epistle of the Number .......................... 33 The mathematical contents of The Epistle of the Number ......................................... 33 Book I ...................................................................................................................... 33 Book II .................................................................................................................... 40 A detailed analysis of Book II .................................................................................... 41 Part I: Proportions and scales .................................................................................... 44 Chapter I: arithmetical methods to find the unknown number ..................... 44 Part II: Restoration and opposition .......................................................................... 49 Chapter I: The basics of algebra .......................................................................... 49 Chapter II: The procedure for the six equations .............................................. 56 Chapter III: Algebraic expressions ...................................................................... 62 Chapter IV: The multiplication of algebraic expressions ................................ 67 Chapter IV: Division ............................................................................................. 72 Part III: Problems of practical nature ....................................................................... 74 Chapter I: Theory and practice for the solution of practical problems ......................................................................................................... 74 Appendix I: The role of zero in The Epistle of The Number ..................................... 87 Appendix II: Traces of algebra in Hebrew before The Epistle of the Number ....... 88 The Book on Mensuration ................................................................................... 89 The Book of Measures .......................................................................................... 90 The Book on Demonstration and Memorization of the Science of Problems Involving Dust Reckoning ........................................................ 90 An anonymous commentary to The Book of the Number ................................... 92 Chapter 4: The Unicum ....................................................................................................... 95 Cambridge University Library MS Heb. Add. 492: codicological and palaeographical notes ......................................................................................... 95 Errors in the unicum ............................................................................................. 97 Diagrams, tables and decorations in the unicum .............................................. 98 Chapter 5: An Edition of The Epistle of the Number .......................................................... 99 Notes to the edition ..................................................................................................... 99 The Hebrew in the edition ......................................................................................... 99 The edition ..................................................................................................................100 Chapter 6: An English Translation of The Epistle of the Number ................................... 257 Notes to the translation ............................................................................................257 The translation............................................................................................................ 257 Chapter 7: A Lexicon of the Mathematical Terms in Book II .................................... 425 Notes to the lexicon .................................................................................................. 425 Bibliography ......................................................................................................................... 459 Primary sources .......................................................................................................... 459 Books ..................................................................................................................... 459 Manuscripts ........................................................................................................... 460

TABLE OF CONTENTS

vii

Secondary Sources ..................................................................................................... 461 Encyclopedias, dictionaries, etc. ........................................................................ 465 Index ..................................................................................................................................... 467

ACKNOWLEDGMENTS Many people have paved my way into the fascinating world of medieval Hebrew and Arabic science. I am most grateful to Leo Corry and Tony Lévy, my doctoral advisors, for their invaluable guidance throughout the composition of my dissertation, out of which this book was born. I am grateful to Cambridge University Library, for allowing me access to the only surviving manuscript of The Epistle of the Number (MS Heb. Add. 492.1). Many thanks to Stefan Reif, Ben Outhwaite, and Godfrey Waller for their advice and assistance. I would also like to thank Benjamin Richler, Abraham David, and Yael Okun from the Institute of Microfilmed Hebrew Manuscripts at the National Library of Israel in Jerusalem, for their kind help during my work on various microfilms. I thank the following people at the CNRS/SPHERE in Paris for everything they have taught me about medieval Arabic and Hebrew science and philosophy: Phillipe Abgrall, Hélène Bellosta (z”l), Pascal Crozet, Gad Freudenthal, Ahmad Hasnaoui, Mehrnaz Katouzian-Safadi, Régis Morelon, and Roshdi Rashed. I am also grateful to my colleagues at the department of Hebrew and Jewish Studies at University College London for helpful advice: Ada Rapoport-Albert, Lily Kahn, Israel Sandman, Kineret Sittig, Sacha Stern, and Nadia Vidro. I thank Judith Olszówy-Schlanger for examining the manuscript of The Epistle of the Number and evaluating its palaeographic aspects as well as her constant support and friendship. Many thanks to Stephen Bacon (z”l), who read and corrected the English in my work, as well as Juliane Lay and Jesús del Prado Plumed for many helpful references. I would also like to express my gratitude to Ruth Glasner and Steven Harvey for their help in solving several difficult translation issues. Many thanks also to Rabbi Dov Berkowici, Bernard Goldstein, Eva Haverkampf, Benjamin Zeev Kedar, Tzvi Langermann, Reimund Leicht, Peter Lipton (z”l), Giuseppe Mandalà, Dov Schwartz, Shlomo Sela, and David Wassserstein, for precious advice. I would also like to thank the following institutions for generous funding: the Sophia Fund for doctoral students in history at Tel Aviv University, the doctoral scholarship from the French government, Yad ha-Nadiv, as well as the research fellowship from Memorial Foundation for Jewish Culture in New York. Finally, I wish to thank my family and friends for their love and support.

ix

FOREWORD My book aims to show how Ibn al-’s famous Arabic text on arithmetic and algebra    -  (A Summary of the operations of Calculation) was transmitted into Hebrew by the polymath Isaac Ibn al-Adab, resulting in the extensive text The Epistle of the Number (  ). My book presents the first edition of The Epistle of the Number, which was composed in Syracuse, Sicily, at the end of the 14th century. The Hebrew Epistle plays a pivotal role in the history of medieval Hebrew mathematics because it is the first known Hebrew treatise which includes extensive algebraic theories and procedures. It is also the first (and last) known version of    - . The Hebrew Epistle exposes novel mathematical vocabulary and enhances our understanding of the linguistic mechanisms which helped create scientific vocabulary in medieval Hebrew. My book also depicts the fascinating figure of Isaac Ibn al-Adab: astronomer, mathematician, poet and exegete – shedding new light on his persona and intellectual activity. Isaac probably left his homeland before the 1391 persecutions in Castile. According to the colophon of The Epistle of the Number, he also spent some time in Muslim land, probably in North Africa, where he studied    - . Later, during a perilous voyage to the Holy Land, Isaac was ship-wrecked in Syracuse. Some of his friends, members of the Jewish community there, asked him to compose a book on mathematics. In response to their request, Isaac decided to translate the Arabic text into Hebrew and adapt it for the Jewish community. This is how The Epistle of the Number came to be. I have endeavoured to present a broad picture of Isaac’s life within the historical and intellectual context of his time. I describe Isaac’s exchanges of knowledge with medieval Jewish scholars such as Samuel Ibn ara, the RiBaSH and Judah bar Asher the Second, who was Isaac’s astronomy teacher. Isaac was mostly known for his astronomical treatises and their high scientific level. He even improved existing astronomical tools, as described in his treatise The Intermediate Instrument (  ). Isaac also wrote about the Jewish calendar and conversion algorithms between the Muslim and Jewish calendars. In addition, he wrote numerous poems, an exegesis of the Passover Haggadah and a book on weights and measures in the Bible. The Epistle of the Number includes an impeccable translation of the    al-  as well as lengthy commentaries and numerous examples as well as theoretical additions and rectifications by Isaac. He explicitly expressed his wish to be understood by his readers and he indeed succeeded in being highly methodical and clear. Not only did he explain the meaning of new mathematical terms, but he xi

xii

THE EPISTLE OF THE NUMBER

may have been the one who coined some of them. He also discussed the etymology of several terms, thereby demonstrating his linguistic sensitivity and creativity. In addition to the abundant philological discussions within, The Epistle of the Number embarks upon interesting analyses – for example, the analogue between the arithmetician, who is expected to apply mathematical operations upon numbers rather than study their nature and the doctor, who is expected to heal, i.e. apply practical medical knowledge, rather than study the quiddity of Man. The Hebrew Epistle bears witness to the multidisciplinary nature of Hebrew scientific works in the Middle Ages as well as the broad expertise of their authors. The Epistle of the Number is a rich text which introduces a wide variety of mathematical themes, both in arithmetic and algebra. In complete congruence with its Arabic source, it is divided into two books. The first book includes the presentation of the place-value decimal system. It elaborates on the basic arithmetical operations: addition, subtraction, multiplication, division and the extraction of the square root applied on integers, fractions, as well as expressible and inexpressible numbers. 1 Numerous algorithms are provided and carefully explained, including the famous ancient Greek method to identify prime numbers, known as Eratosthenes’ sieve. The second book in the Hebrew Epistle presents rules and procedures which “allow us to find the value of the sought unknown from the given known”: the rule of three, the rule of double false position and algebraic procedures, such as the solution of equations of the first and second degree. The domain of definition of the arithmetical operations defined in the first book is extended in the second book from numbers to various algebraic species: roots, squares and cubes, thus creating algebraic expressions, the very remote “ancestors” of polynomials. After a rigorous analysis of the algebraic theories, we find a series of problems of verbal reasoning: distance-time-velocity calculations, charity distribution and the purchase of a horse by several associates. I have composed a commentary on the mathematical themes in The Epistle of the Number focussing on Book II. I have cautiously used modern mathematical notation to describe the various theories, which were originally written in prose. To the best of our knowledge, there is only one surviving copy of The Epistle of the Number, a mid-16th century copy from Constantinople. The manuscript is kept at Cambridge University Library, and its shelf-mark is MS Heb. Add. 492.1 (folios 1v– 38v). 2 I present the palaeographical and codicological features of the unicum and the codex in which it is found, followed by an edition of The Epistle of the Number, with notes and a translation into English. To summarize: my book aspires to show the various intellectual facets in Isaac Ibn al-Adab’s work, in terms of text and context, focussing on The Epistle of the

1

In modern mathematical language: rational and irrational numbers, respectively. See Stefan C. Reif, Hebrew Manuscripts at Cambridge University Library, A description and introduction, (Cambridge, 1997), p. 590. 2

FOREWORD

xiii

Number. Apart from being a tract on a novel subject on the medieval Hebrew mathematical bookshelf, the Epistle is manifold testimony of the transmission of Arabic science to Jewish circles in Christian Syracuse at the end of the 14th century and reflects the scientific activity of the Jews there, of which still relatively little is known. My book also aims to fill a gap in the exposure of medieval Hebrew mathematical texts 3 as only a small part of Hebrew mathematical materials from the Middle Ages have been edited so far, with an even smaller number of tracts which were also translated or commentated upon. This book will be of interest to scholars in the fields of Jewish Studies, Hebrew philology, medieval studies, mathematics, history of science, and, in particular, history of medieval Hebrew and Arabic mathematics.

3 Furthermore, my book comprises the only extant study of The Epistle of the Number, preceded only by one article on selected aspects of the text by Tony Lévy as well as five short articles which I have written, which include materials from my doctoral dissertation and thus, from this book. See Lévy, Tony, ‘L’algèbre arabe dans les textes hébraïques (I), Un Ouvrage Inédit D’Isaac ben Solomon al-     Arabic Sciences and Philosophy, 13, (2003), 269–301 and Ilana Wartenberg, ‘The Epistle of the Number: An Episode of Algebra in Hebrew’, Zutot: Perspective on Jewish Culture 5 (2008), 95–101; ead., ‘Iggeret haMispar: by Isaac ben Solomon Ibn al-Ahdab (Sicily, 14th century) (Part I: The Author)’, Judaica 64 (2008), Heft 1, 18–36; ead., ‘Iggeret ha-Mispar: by Isaac ben Solomon Ibn alAhdab (Sicily, 14th century) (Part II: The Text)’, Judaica 64, Heft 2/3 (2008), 149–161; ead., ‘The Epistle of the Number: The Diffusion of Arabic Mathematics in Medieval Europe’, Circolazione Dei Saperi Nel Mediterraneo: Filosofia E Scienze (Secoli IX–XVII), Circulation des Savoirs Autour de la Méditerranée: Philosophie Et Sciences (IXe–XVIIe Siècle): Atti Del VII Colloquio Internazionale Della Société Internationale d’Histoire des Sciences Et de la Philosophie Arabes Et Islamiques: Firenze, 16–28 Febbraio 2006, Ed. Graziella Federici-Vescovini and Ahmad Hasnaoui, pp. 101–110 (Edizioni Cadmo, 2013) and ead., ‘The Naissance of the Medieval Hebrew Mathematical Language as manifest in Ibn al-Ahdab’s Iggeret ha-Mispar’, in A Universal Art: Hebrew grammar across disciplines and faiths, Ed. Nadia Vidro, Irene Zwiep, Judith Olszowy-Schlanger, (Brill, 2014), pp. 117-131.

HEBREW ABBREVIATIONS

" "  \\"   "    "  3  .  '

 '  '

     :   .

   "

 "

 "

 " "

 "  "    "  "

 " "            

  1       "  " " 2   " "  " "

1

“And to him that hath no might He increaseth strength” [Isaiah 40:29]. I have not been able to decipher this abbreviation, which appears on folio 17r. 3 Often words at the end of a line were shortened, i.e. their end was omitted, for justification purposes, and they are easily decipherable for Hebrew readers. 2

xv

HEBREW AND ARABIC TRANSLITERATION I. Hebrew has been transliterated in accordance with the following system: 1 Consonants r š s t

 ! % 



y k kh l m n s  p f  q

#       (   

 b v g d h w u o z  

 $    

 & )   *

Vowels Vowel length is not represented. Vocalic šewa is not transliterated The transcription of aef vowels is identical with that of their full counterparts. qama, pata = a ere, segol = e iriq = i olam, qama qaan = o shuruq, qubbu = u

1

However, for standardized English spellings of terms or names (e.g. RiBaSH) I have not altered their transliteration, even if they do not correspond to the system I use elsewhere. The same goes for spelling used by other authors in the bibliography and for Arabic terms and names.

xvii

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THE EPISTLE OF THE NUMBER

II. Arabic has been transliterated in accordance with the following system:   !  # f q k l m n h w y Long Vowels

= =% =& Short Vowels fata = a kasra = i amma = u


  

       

 b t " j  kh d $ r z s š 

   

       

CHAPTER 1: IBN AL-ADAB INTRODUCTION In 1899, the father of modern Jewish bibliography and one of the founders of modern Jewish scholarship (Wissenschaft des Judentums), Moritz Steinschneider (1816–1907), wrote in his seminal work Mathematik bei den Juden, that Isaac ben Solomon Ibn al-Adab “has not won the interest of scholars of Hebrew literature in the way that he deserves. They only mention him every now and then”. 1 This statement is no longer true today, as research has revealed new facets of Isaac’s scientific, literary and exegetical works. However, to date no thorough study of his mathematical work has ever been made and this books aims to fill this gap. This chapter presents the highlights of Isaac’s life and ensemble d’oeuvres, as well as his reading sources and intellectual and familial milieu, briefly describing the historical circumstances in Castile and Sicily during Isaac’s lifetime and Isaac’s whereabouts during the persecutions of 1391 – a key traumatic event to the Jewish communities in the Iberian Peninsula and Italy. Finally, the relevant contextual elements behind the composition of The Epistle of the Number are examined, shedding light on the cultural and religious setting in which Isaac was active.

ISAAC’S LIFE AND TIMES Isaac ben Solomon Ibn al-Adab ben addiq ha-Sefardi (i.e. the Spaniard, Sephardi) was born in Castile between 1340 and 1350 and passed away sometime between 1429 and 1433. In a poem of admiration dating from 1377, which Isaac wrote to the Samuel Ibn ara, Isaac refers to himself as “young”. 2 1429 is the last known year in which it is certain that Isaac was alive. It was in this year that his student, Nissim      -Faraj, copied one of Isaac’s texts in which he sought God’s blessing to prolong his teacher’s life. But in a treatise copied in 1433 by the same scribe, Isaac is referred to “in blessed memory” (z”l). 3 1

“Isak Alchadib hat bisher nicht das verdiente Interesse der Forscher auf dem Gebiete der jüdischen Literatur gefunden, die nur obenhin oder gelegentlich seiner erwähnen…“. See Moritz Steinschneider, Mathematik bei den Juden, (Hildesheim, 1964), p. 167. 2 Dov Schwartz, The religious philosophy of Samuel Ibn ara, (in Hebrew), (Ph.D. dissertation, Bar-Ilan University, Ramat-Gan, 1989), pp. 5–7. 3 The Intermediate Instrument, Munich, Bayerische Staatsbibliothek, MS Cod. Hebr. 246/9, fol. 67v (IMHM 1102), A Poem on Esther, Munich, Bayerische Staatsbibliothek, MS Cod.

1

2

THE EPISTLE OF THE NUMBER

Isaac was trained in Castile by the astronomer Judah ben Asher the Second, the great-grandson of Rabbi Asher ben Yeiel (the RoSH). 4 At some point, probably before the 1391 persecutions, Isaac left Castile. As we read in the colophon of The Epistle of the Number, Isaac also lived in a Muslim land, either in the Muslim part of the Iberian Peninsula (Granada) or, more likely, in North Africa. In his astronomical treatise A Precious Instrument () Isaac tells us that in 1396 he was in Syracuse on the island of Sicily. In the year [5]156 to the Creation 5 I am in Syracuse, which is in the Island of cold Sicily, and God is in front of me. I have invented an instrument, which is very easy to use; also, with it one can easily establish the positions of the planets… 6

In the introduction to The Epistle of the Number we read of Isaac’s arrival in Syracuse: Later, while I was going through the sea to the Holy Land, may it be built and existent in our days, the waves of the sea were threatening us, deep calleth unto deep, 7 our soul is bowed down to dust. 8 Observing us from His Seat, The Observer of His world in general and in particular, may His name be praised, exalted and raised, calmed the stormy waves of the sea and brought us all safe to the glorious town of Syracuse in Sicily. [The Epistle of the Number, Cambridge, University Library, MS Heb. Add. 492.1, fol. 1v]

Isaac recounts his maritime journey to the Holy Land, a journey interrupted by a vicious storm, leaving him stranded in Syracuse, Sicily. However, this account should be taken with a pinch of salt since this type of story, involving elements of a

Hebr. 246/19, fol. 254r (IMHM 01102) and An article on the definition of things, Munich, Bayerische Staatsbibliothek, MS Cod. Hebr. 246/8, fol. 67r (IMHM 01102). The number in the round brackets refers to the microfilm or microfiche number (the latter is explicitly marked as ‘Fiche’) at the Institute of Microfilmed Hebrew Manuscripts (=IMHM) at the National Library of Israel in Jerusalem. 4 Leopold Zunz, Zur Geschichte und Literatur, (Berlin, 1874); reprint (Berlin, 1919), p. 423, Alfred Freimann, ‘Die Ascheriden 1267–1391’, in Jahrbuch der jüdisch-literarischen Gesellschaft, 13, (Berlin, 1919), pp. 155–156. 5 I.e. 1395/6 CE. 6 A Precious Instrument, London, British Library, MS Or. 2806, fol. 11r (IMHM 06385). According to the entry on top of the folio, we learn that the owner of the manuscript is not other than (the young) Mordekhai Comtino, 15th century Turkish Talmudist, astronomer and mathematician. He is not the scribe of the text, since his entry is in a substantially different handwriting than the one we see in the text. 7 Psalm 42:7. 8 In Psalm 44:25 we find the semantically equivalent    “our soul is bowed down to the dust…” Isaac slightly adapted the verse according to rules known as melia  .

IBN AL-A'DAB

3

voyage to the Holy Land, a storm and a shipwreck, 9 belongs to a literary topos typical of Hebrew literature in the Middle Ages and is not necessarily a reflection of reality. 10 Isaac first stayed in Syracuse, where he composed The Epistle of the Number and the astronomical tract A Precious Instrument. The last evidence of Isaac’s location can be found in his astronomical tract The Intermediate Instrument, from which we learn that Isaac was in Palermo in 1426. 11

THE HISTORICAL CIRCUMSTANCES A full description of the historical events which directly influenced the Jews in Castile and Aragon in the fourteenth century is outside the scope of this book. I shall only point to the highlights of the complex circumstances which belong to Isaac’s lifetime – a tumultuous period for Jews both in Castile and Aragon. 12 Castile The fourteenth century marked the beginning of the destruction of Castilian Jewry. Between 1348 and 1350, Jews were accused of causing the Black Plague, and as a consequence, they were persecuted. Furthermore, the middle of the fourteenth century was characterized by a bloody civil war in Castile, which bore tragic consequences for the Jews as well as the wider population. Pedro, the legitimate son of Alfonso the Eleventh, ruled Castile with the help of Jews, the most famous of whom was Don Samuel ha-Levi. In 1366, the son of Alfonso’s mistress, Enrico the Second, rebelled against his brother’s reign, denouncing Pedro as the son of a Jew. In 1369, Pedro was killed and Enrico the Second became the supreme authority in Castile. Many Jews were massacred and numerous Talmudic academies were destroyed, as one can read in a letter by Rabbi ben Jacob Lattes. The Jews also had to pay heavy forfeits to the new king, creating an even greater burden. The 9

Although no boat or ship is explicitly mentioned by Isaac and in the absence of medieval submarines, it is hard to think of other means to travel around the Mediterranean Sea back then. 10 Tova Rosen, ‘The Hebrew Mariner and the Beast’, Mediterranean Historical Review, 238–244. 11 Giuseppe Mandalà has made an important discovery; he identified Isaac Ibn al as Gaudio Alachadeb, a known arbitrary (borer), notaries Iudeorum, and probably a spiritual judge (dayan) in Palermo. Mandalà researched documents showing that Isaac/Gaudio was already living in Palermo in 1418. I wish to thank Giuseppe Mandalà for sending me his article, which also sheds new light on the life of Isaac, his family, the Sicilian Jewish community as well as the historical context in Sicily. See Giuseppe Mandalà, Da Toledo a Palermo: Yiaq ben Šelomoh ibn al-Adab in Sicilia (ca. 1395/96 – 1431), Flavio Mitridate mediatore fra culture nel contesto dell’ebraismo siciliano del XV secolo, Atti del convegno internazionale di studi, Caltabellotta (Agrigento), 30 giugno-1 luglio 2008 - Palermo, ed. Mauro Perani and Giacomo Corazzol. (Palermo, 2012), pp. 1–16. 12 This section presents succinct highlights from Yitzhak Baer, Toldot ha-Yehudim bi Sefarad ha-Notsrit, (Tel-Aviv, 1959), pp. 179–363.

4

THE EPISTLE OF THE NUMBER

humiliation of the Jews was further extended when they were forced to wear a distinctive Jewish mark. Enrico’s son, John the First, ruled between 1379 and 1390. During his reign, various anti-Jewish laws were enforced, while at the same time several Castilian Jews held high rank in the Royal Courts. John the First of Castile died in 1390, before his heir was mature enough to inherit the crown. The custodians appointed by the late king were unable to curb the rebelling forces in Castile. On June 4th, 1391, these forces set fire to Jewish homes in Seville, massacring part of the population, coercing others to convert to Christianity and selling some of them to the Muslims. This was to become the pivotal and most traumatic event for Iberian Jews in the fourteenth century. Many Jews chose martyrdom, as related in 'asdai Crescas’ letter to Avignon’s rabbis, in which he describes the martyrdom of Judah ben Asher the Second, Isaac’s teacher of astronomy. Synagogues were converted into churches and Jewish neighbourhoods were occupied by Christians, leading to the complete destruction of Jewish communities, such as the one in Valencia. Aragon Historians deem the fate of Jews in Aragon in the fourteenth century to have been better than that of the Jews in Castile, although it was not completely devoid of violent acts against Jews. Sicily, which was part of Aragon’s domain, enjoyed an even greater peace in comparison to the rest of Aragon. Pere the Third ruled Aragon between 1336 and 1387. His reign was relatively calm and was characterized by the reconstruction of the Aragonese Jewish community. In general, Pere the Third protected the rights of the Jews and allowed their commercial life to flourish. In particular, he gave them permission to import fabrics from England and Flanders. Both Pere II and his son John the First, were interested in astronomy and had Jewish scholars in their court. 13 It is very difficult to establish a realistic picture of the conditions of the Jews in Aragon in spite of the existing documents written in Hebrew by Jewish merchants and doctors, some preserved in archives in Pamplona. These documents indicate that Jews were secure both financially and socially. Pere the Third seems to have made an effort to be an enlightened king. Despite sporadic instances of malicious suspicion of the Jews, he was served by Jewish doctors, astronomers, financial advisors and translators. He is also known to have given religious autonomy to the Jews in regard to capital cases. Profound hatred against the Jews was rooted both in Castile and Aragon, but its manifestation was more extensive and overt in Castile. Baer explains that this phenomenon was related to the popular belief that Jews had desecrated the host and

13

Chabás, José and Goldstein, Bernard R., ‘Isaac Ibn al-'adib and Flavius Mithridates: The Diffusion of an Iberian Astronomical Tradition in the Late Middle Ages’, Journal for the History of Astronomy, xxxvii, (2006), 147.

IBN AL-A'DAB

5

that Jewish religious writings, such as Maimonides’ – the RaMBaM’s – Mishneh Torah, contained anti-Christian sayings. In 1387, Pedro’s son, John the First, assumed power in Aragon until 1396. Once becoming king, he dismissed all libellous stories against the Jews, even though he had himself promulgated similar allegations earlier. Like his father, he maintained good relationships with the Jews of Aragon, in particular with 'asdai Crescas. However, Pedro was a weak king. In July 1391, the first rumours of persecutions in Castile had reached Aragon and caused uproars. John the First published orders stating that the Jews were considered property of the King and that the clerks must protect them. He even sent chevaliers and soldiers to protect the Jews in various cities in Aragon but at the same time did not hesitate to steal the property of the Jews whose lives had been lost in the persecutions. Violence persisted for several years. In 1392, Jews in Sicily were confined to their quarters. There were also persecutions in the summer of that year, which ended shortly after Martin the Fifth of Aragon ordered the punishment of the perpetrators, thus protecting the Jewish community in Sicily. The Jewish community in Sicily During the fourteenth century, Sicily was under Aragonese Christian rule. Jews and Muslims living in Christian Spain were grouped in local Aljamas ruled by lay as well as religious leadership. Long after the end of Muslim rule in Sicily in 1072, we know that Sicily still preserved strong linguistic and cultural Arabic imprints. Arabic was still in widespread use among Sicilian Jews in the fourteenth century. Manuscripts as well as commercial and legal certificates indicate the usage of Hebrew, Arabic, Catalan and Aragonese, although the quality of the Arabic was at times poor. 14 Arabic culture still influenced Jews in Aragon, as manifest in poetry and artistic work, songs and even Jewish food and language of the time. For example, Jews referred to Sunday al-aad 15 and not domingo. Commercial documents suggest that transactions between Muslims and Jews were common. In the thirteenth century, legislation against Jews converting to Islam was enacted both in Aragon and Castile, suggesting that conversion to Islam was a common phenomenon among the Jews there. In general, the Jews living in Castile and Aragon bore a strong affinity to Muslim culture and religion. 16 This affinity is corroborated in Isaac’s writings: the knowledge of Arabic, at least by some members of the Jewish community in Syracuse, is alluded to in The Epistle of the Number. In the introduction to the Epistle, Isaac clearly expresses his concern that some Sicilian Jews who understand Arabic might find out that some passages from the Arabic source are missing from his Hebrew translation. 14

Henri Bresc, Arabes de langue, Juifs de religion, (Paris, 2001), pp. 46–50. Or alhad, which is still used in Judaeo-Spanish nowadays. 16 Eleazar Gutwirth, ‘Hispano-Jewish Attitudes to the Moors in the fifteenth Century’, Sefarad, (1989), 245–250. 15

6

THE EPISTLE OF THE NUMBER

I saw fit to omit a few parts from the Epistle, since, in my view, they are of no use. I shall point out to those places as I reach them so that someone who will read the Epistle in the Arabic language will not be able to catch me in flagrante delicto. [fol. 1v]

Sicilian Jews were successfully involved in the local economy. They practiced various professions: tax-collectors, financiers and merchants. Surviving certificates indicate that many Sicilian Jews in the fourteenth century were in possession of small plots of land. There was a Jewish Sicilian bourgeoisie mind-set in regard to housing and concern with domestic comfort – Jews contributed to the commercial developments of Sicilian towns and enjoyed a great freedom of movement. 17 It is difficult to say much about the interest of fourteenth-century Sicilian Jews in mathematics per se. Isaac’s Epistle is the only known Hebrew mathematical book to have been written in Sicily at the time. A Hebrew commentary on Menelaus’ De Sphaera, attributed to Jeremiah ha-Kohen of Palermo, is in actuality a treatise on the sundial with little mathematical value. 18 Furthermore, the Sicilian Jewish communities did not enjoy a high reputation for Talmudic knowledge. The casuistic literature of the medieval Rabbinic Responsa includes few references to Sicilian Jewry. 19 In the introduction to The Epistle of the Number Isaac describes some members of the Jewish community in Syracuse who were interested both in Torah and in mathematics. 20 There, I found many honourable men of high rank, who busied themselves with the Torah and the precepts. Amongst them, their seed is established in their sight

17

Idem, ‘Widows, Artisans and the Issues of Life: Hispano-Jewish Bourgeois Ideology’, Iberia and Beyond, Hispanic Jews between Cultures, (1992), pp. 151–152. 18 The attribution derives from a confusion in the description of Vatican MS Ebr. 379 in S. Assemanus and J. Assemanus, Bibliothecae Apostolicae Vaticanae…I: Codices ebraicos et samaritanos (Rome, 1756), p. 354, where several passages on folios 67–74 are described as a Commentarius de Tractatum de Sphaera by Jeremiah ha-Kohen of Palermo. This mistake is repeated by C. Roth, “Jewish Intellectual Life in Medieval Sicily,” The Jewish Quarterly Review 47 (1957), p. 324, who relies on L. Zunz, Zur Geschichte. M. Steinschneider, Die hebraeischen Übersetzungen des Mittelalters (Berlin, 1893), p. 542, explained the source of the confusion and showed that the text by Jeremiah, written in 1486, is actually on the sundial. I have examined this text and it is of little value for the history of Hebrew mathematics in Sicily. In any case, it is known that astronomical activity was more abundant than mathematical one in Sicily (and elsewhere), as can be testified also by the number of astronomical works by Isaac. I wish to thank Steven Harvey for having pointed this confusion out to me. 19 Cecil Roth, ‘Jewish Intellectual Life in Medieval Sicily’, The Jewish Quarterly Review, *

 + @Q X\X–334. 20 Astronomical activity was more abundant than mathematics in Sicily, as can be testified from al-Adab’s known works in the field. This is discussed later in this chapter.

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with them. 21 They were educated, lovely young men, whose friendship I enjoyed. I chose their company; some of them studied with me the teachings of our Holy Torah. In their free time, some of them were studying the Science of the Number. One day, they asked me to compose for them a short book in this science, as comprehensive in all matters as possible. [fol. 1v]

PEOPLE IN DIRECT CONTACT WITH ISAAC The RiBaSH (Valencia 1326–Algiers 1408) Rabbi Isaac bar Sheshet Perfet, or Barfat, the RiBaSH, was a prominent Jewish scholar of noble descent. He was born in Barcelona, and in 1372, he moved to Saragossa, where he was leading the local Jewish community from 1385 until 1391. Regarding foreign sciences he stated: One must not discuss the laws of the Bible and its precepts using the wisdom of Nature and medicine, because if we were to believe their sayings, then the Bible would not come from Heaven, God forbid… we rely upon our wise men, of blessed memory, even if they tell us that right is left… We should not believe the Greek scholars and the Muslims, who only talk out of their own assumptions and experience. 22

During the persecutions of 1391, the RiBaSH was one of the few who were able to escape from Valencia. Regarding the relationship between the RiBash and Isaac, we only know that in a responsum addressed to Judah ben Asher the Second, the RiBaSH stated that he had met Isaac in Castile before the persecutions. He refers to “three quires, which your student, al-Adab, laid here, may God save him; he wrote them for me to send to you”. The content of these notebooks is unknown. 23 The RoSH’s family: Judah ben Asher the Second Judah ben Asher the Second was the great-grandson of the RoSH, i.e. Rabbi Asher ben Yeiel, the chief Rabbi of Toledo. At the beginning of the fourteenth century, the RoSH was invited to Toledo by the local Jewish community. Despite the fact that the RoSH originated from an Ashkenazi milieu (Germany), he was highly influential in the life of the Sephardi community of Toledo and his religious rulings were fully respected. After his death, one of his grandsons, Judah the First, succeeded him. Then, after Judah’s death in 1349, the RoSH’s great-grandson, Judah ben Asher the Second from Burgos, became Rabbi in Toledo. The latter wrote a book on astronomy and was Isaac’s teacher on the subject. During the persecutions 21

Job 21:8. Baer, Toldot ha-Yehudim, p. 271. 23 The Responsa project, (in Hebrew), Bar-Ilan University, (Ramat-Gan, 1972–2002), Responsa 240 (CD ROM). 22

8

THE EPISTLE OF THE NUMBER

of 1391, Judah chose martyrdom in Toledo, as did many of his friends and family members. 24 Samuel Ibn ara (Valencia, second half of the 14th century) Samuel Ibn ara was active in Castile between 1360 and 1380. In his treatises on philosophy he wrote about the theory of divine attributes, the Creation of the World, morality (based on Maimonidean theories), divine Providence, and the source of the soul. Ibn ara wrote two exegetical works: The Source of Life and A Perfect Beauty. The Source of Life comments on Ibn Ezra’s biblical exegesis. A Perfect Beauty is a commentary on tales of our Sages of Blessed Memory and their Midrashim, i.e. homiletic interpretations. Ibn ara’s period is dominated by Averroistic rationalistic theories, but in many respects Ibn ara’s doctrine is neoPlatonic. Ibn ara was trying to create a synthesis between rationalism and NeoPlatonism, interwoven with strong elements of astrology and astral magic, which endows human beings the capacity to change nature. 25 Around 1377, after having studied Ibn ara’s works, Isaac sent the author a letter of admiration written in verse, at the end of which he refers to himself as “Ibn al-Adab the young”. This letter is the only known correspondence between Isaac and Ibn ara. The latter was also known by the name Ibn/Ben Senne, but its nonvocalised form in Hebrew SNH () has misled historians into believing that Isaac             !   matter of fact, the Jewish name Ibn Seneh derives from the translation of the Spanish name ara, a thorn bush, into the Hebrew Seneh. This letter sheds some light on Isaac’s philosophical and exegetical background since in it he expresses his profound respect for Ben Senne, whose wisdom and splendour had superseded all men of his generation. Isaac prayed for the welfare of Ibn ara and expressed his hope that God would prolong his days. Isaac was familiar with medical knowledge, at least on some level. In The Epistle of the Number,   "#Canon of Medicine (

 ) in Book I, when he discusses ratios of a series of 4 numbers. 26 32F

    

Zaraia Ibn Danon was a poet with whom Isaac had a polemic exchange of poems, the last of which Isaac wrote on Ibn Danon’s tomb, immediately after the latter’s death. 27

24

Baer, Toldot ha-Yehudim, p. 285. Schwartz, The religious philosophy, pp. 13–18. 26 The Epistle, fol. 3r. In 1481, the five books of the Canon of Medicine in Hebrew translation were sold together with other books to Jews in Polizzi by Isaac’s grandson Gaudio the Younger – these books had probably been in the possession of the grandfather Isaac/Gaudio. See Mandalà, ‘Da Toledo a Palermo’, 12. 27 Ora Raanan, Poems of Rabbi Ishak ben Shlomo ben addik al-Adab, (Lod, 1988), p. 26. 25

IBN AL-A'DAB

9

Isaac’s family Ibn al-Adab literally means in Arabic ‘the son of the hunchback’. Isaac’s surname appears in the manuscripts under several forms, for example: al-Adab, al-'adib, al'adab, and al-Khadib. Several members of the al-Adab family are known to have lived in Toledo: Solomon bar Samuel Ibn al-Adab, as we learn from an anonymous inscription of the names on Jewish tombs. According to the memorial stone in the cemetery of Toledo, Solomon Ibn al-Adab died in 1349, during the Black Plague. 28 What we know for certain is that Isaac had (at least) three sons: Abraham, addiq, and Jacob. The latter had a son named Gaudio (a clear case of papponomy), who was a doctor. Isaac and some of his descendants played an important public, intellectual as well as economical role in the Sicilian Jewish community during the fifteenth century. 29 Jacob Jacob was Isaac’s youngest son. He wrote: “I am the youngest at my father’s home”. 30 Jacob copied some of his father’s poems and wrote commentaries on The Paved Way (   ). He also elaborated on his father’s astronomical tables, in response to a friend’s request. 31 Jacob is also known to have predicted the solar eclipse of 1463. In a commentary on The Paved Way, Jacob writes: 37F

In our times our brain is meagre and meek, and we have no strength to enter the ample ascensions established for us by the first scholars in the Temples of Wisdom many years ago, the wisdom of astronomers and astrologers. Due to the work, labour and toil of our time, our knowledge has diminished such that when we find commentated books accompanied by commentated exegesis with the paucity of our mind we are not capable of understanding these matters, in particular, the books on astronomical tables which involve calculations and numbers, unless there are numerical examples and commentaries. Therefore, I, the youngest at my father’s home, have consented to the request of a beloved

28

Yosef S. Spiegel, Haggadah shel pessach, pessach dorot, dinei leil ha-seder u-perush ‘al hahaggadah le-    - ! - "! #-Si$ilia ba-me’a ha-shniya laelef ha-, (The Passover Haggadah commentated by al- ^_` \{{{, p. 1. Could the mentioned Solomon have been Isaac’s father? If so, then Isaac was born in 1350 at the latest. Another possible family connection: in his commentary on the Passover Haggadah, Isaac refers to a certain Judah Krishef as his grandfather. But we know of a Judah Krishef who was one the RoSH’s sons-in-law. In his commentary on the Passover Haggadah, Isaac refers to a certain Judah Krishef as his grandfather. If both Judahs are the same person, then Isaac must have been the RoSH’s great-grandson. See further idem, Haggadah shel pessach, pp. 9–10. 29 For details see Mandalà, ‘Da Toledo a Palermo’, 9–16. 30 The Paved Way, London, British Library, MS Or. 2806, fol. 20v (IMHM 06385). 31 The Paved Way, Paris, Biblothèque Nationale, MS héb. 1047/18, fol. 172r (IMHM 14650).

10

THE EPISTLE OF THE NUMBER

friend, who wishes to learn the tables of The Paved Way, composed by my master, my father, of blessed memory. My friend has asked me to give him an example from the gentiles’ tables, which will facilitate the calculation 32 of an eclipse and the opposition of the luminaries. 33 For the sake of his friendship and in order to fulfil his will and respond to his request, I have written this example for him and all those who wish to study gentile tables with ease. I ask the reader to forgive me if he finds any mistakes, may God save us from errors. I shall start, with God’s help, may His name be blessed and exalted… 34

 ! !! Abraham and addiq were the elder brothers of Jacob, for whom Isaac wrote affectionate poems on their wedding day. He advises them to behave morally and how to choose the right path in life. As can be read in his poems, despite Isaac’s extensive scientific activities, like all medieval Jewish scholars he attached foremost importance to religious studies. According to Isaac’s poems, science is of secondary importance to, in particular, foreign sciences. One should distance oneself from it if it contradicts the Jewish sources. Isaac’s descendants Some scholars are identified as Isaac’s descendants, such as Rabbi Abraham bar Solomon al-'adab, who was a religious judge in Corfu in 1530. Other possible descendants up to the eighteenth century are to be found in Fez, Venice, and different towns of today’s Greece as well as in Tiberias. 35

WHERE WAS ISAAC IN 1391? It is not clear where Isaac was during the persecutions of 1391, as none of his writings mention these tragic events. The strongest term he uses against Christians is ‘Christian exile’ (     ), which appears in his commentary on the Passover Haggadah; an expression which seems to be a rather general term designating the Reconquista than the persecutions themselves. However, Shirman is strongly

The text says  , lit. ‘the extraction of’. The text says       , but one would expect to find ‘conjunction and opposition of the luminaries’, which are found in the astronomical tables. An eclipse can be either solar (at conjunction, in the beginning of the lunar month) or lunar (at opposition, in the middle of the lunar month) and thus, there seems to be a mismatch of the semantic categories of the used terms. Perhaps this was an imprecise, popular manner to describe eclipses. 34 The Paved Way, London, British Library, MS Heb. Add. 26921, fol. 20v (IMHM 05469). 35 Zunz, Zur Geschichte, pp. 423–424 and Spiegel, Haggadah shel pessach, p. 9. 32 33

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convinced that Isaac fled the persecutions in Castile. 36 It is surprising that no one has ever considered the possibility that Isaac was spared these horrible events, especially since no counter evidence for this option exists. A careful reading of Isaac’s poems does not reveal any hint of the persecutions, but perhaps the poems that reached us date from before 1391. However, none of his other works – whether exegetical or astronomical – written after 1391, bear any indication of these events and in particular of the atrocious death of Isaac’s teacher, Judah ben Asher the Second. Several hypotheses are possible: perhaps the works in which these events were mentioned were lost. It is also possible that the scribe was confined by censorship and hence, could not copy the account of such horrible events, but could this have been the case of all the surviving copies of Isaac’s post$%&$# Unfortunately, at the moment I am not able to provide any details regarding Isaac’s whereabouts in 1391. He may have still been in Castile and miraculously escaped the persecutions. He may have already been in Muslim lands learning mathematics with Muslim scholars or already in Sicily. In any case, I have reason to believe that in 1391, Isaac had long left Castile. Assuming no censorship considerations are valid, I would argue that had Isaac left Castile after the persecutions, one would have expected the introduction of The Epistle of the Number, probably written after 1391, to include a reference to these horrors, in particular because Isaac’s own teacher chose martyrdom in those events. It seems strange that he would tell us about his stay in Muslim lands and the voyage to the Holy Land which was interrupted by a vicious storm, whereas the latter event pales when compared to the persecution. Hence, although I cannot provide a definite answer to Isaac’s location in 1391, I hope to raise the readers’ attention to the possibility that Isaac may have not present, or even knowledgeable of the 1391 persecutions. New evidence may however emerge in the future, which will shed light on this matter. Isaac’s reading sources The following list incorporates Isaac’s identifiable reading sources: x Ibn al-’s Talkh A ml al-isb (Compendium on the operations of Calculation), its author’s commentary Raf al-i (Unveiler of the Veil) and possibly other commentaries. 37 x Within The Epistle of the Number, in his discussion that one is not a number, Isaac also mentions Ibn Rushd’s commentary. His formulation, however, is very different from other Hebrew versions we know of. 38

36 Shirman, '|` ha-Shirah ha- Ivrit bi-Sefarad uve-Provans: mivar shirim ve-sipurim meorazim, (The Hebrew Poetry in Spain and in Provence), (Jerusalem, 1954), p. 582. 37 See details in chapter 2, pp. 25–28. 38 The Epistle, fol. 1v.

12

x

x

THE EPISTLE OF THE NUMBER

Euclid’s Elements: Isaac must have been familiar with the contents of this book, either with one of its translations or commentaries either in Arabic or Hebrew, since in The Epistle of the Number he cites two common notions from The Elements. 39 However, the linguistic discrepancy between the Hebrew version given in The Epistle of the Number and the other Hebrew translations of Euclid which are known to us, strongly hints that Isaac may have not read the Hebrew translations of The Elements. He probably studied one of its Arabic translations during his stay in a Muslim country. Abraham Ibn Ezra’s 40 (Tudela, c. 1089 – c. 1167) work(s): Isaac refers to Ibn Ezra’s biblical exegesis in his Wedge of Gold (  ), a book on weights and measures in the Bible (described further on). It is possible that Isaac also knew Ibn Ezra’s mathematical works such as Book of the One (  ) and Book of the Number () since these books were in wide circulation at the time. In the Epistle Isaac mentions the well-known definition of multiplication in the Holy language, by which he probably refers to Ibn Ezra’s definition. 41  " Canon of Medicine (

). Isaac explicitly mentions it when he compares the ratio between any four numbers and the four humours. 42 Aristotle’s Physics, or a commentary thereof: Isaac explicitly mentions the four causes. 43 Also, Isaac probably read the Organon or any of its commentaries since he discussed the division of discrete and continuous quantities. 44 47F

x

48F

x

The work of the following scholars is referred to in Isaac’s commentary to the Passover Hagaddah: 45 x Maimonides’ Guide for the Perplexed (  ). x RaSHi’s exegesis to the Bible. The following medieval astronomers’ work influenced Isaac’s writing: x Immanuel Bonfils ben Jacob (c. 1350), active in Tarascon: In 1365 Bonfils wrote an astronomical work Six Wings ( ), which became popular and was translated into Latin in 1406. This book is believed to have motivated Isaac to improve the quadrant. 46 52F

39

Ibid, fol. 28r. Although  should have been transliterated as Ezra, Ezra has become the Standard English transliteration. 41 The Epistle, fol. 10r. 42 Ibid, fol. 3r and also see footnote 26 above 43 Ibid, fol. 6v. 44 Ibid, fol. 13v. 45 Spiegel, Haggadah shel pessach, pp. 24–25. 46 See section on The Intermediate Instrument. 40

IBN AL-A'DAB

x

13

Ibn ar-'** ! $+(c. 858 – c. 929) from Harran who was active in ar-Raqqah and Ibn al-?twelfth century, al-Andalus). 47

ISAAC’S WORKS Altogether, there are over eighty manuscripts which refer to Isaac’s works, scattered in tens of libraries around the world. The richness of the domains covered designates Isaac as a polymath in various fields, including astronomy, mathematics, poetry, philosophy, and exegesis. Mathematics The Epistle of the Number, the kernel of this book, is Isaac’s only known purely mathematical text. As mentioned earlier, it is the first (and only) known version of Ibn al-’s renowned Talkh A ml al-isb. It includes a perfect translation of the Arabic source as well as long elaborations. The Epistle of the Number plays a special role in the history of medieval Hebrew mathematics because not only is it the first known Hebrew treatise to include extensive algebraic theories and procedures, but it also exposes a rich novel mathematical vocabulary. 48 The Wedge of Gold (  ) is an exegetical work of mathematical nature in which Isaac comments on the measures and weights mentioned in the Bible. In the course of time, this book was erroneously considered to be lost and then it was mistakenly believed to have been published in Venice in 1552. In this tract, Isaac builds a model of the tabernacle and explores numismatics. 49 This treatise involves many area calculations and hence, apart from being an exegetical work, The Wedge of Gold can be considered of some, albeit marginal, mathematical nature as well. The 5F

47

Spiegel, Haggadah shel pessach, p. 14 and José Chabás and Bernard R. Goldstein, Bernard R., Flavius Mithridates, 148. An explicit reference to Ibn ar-}~~`   -  & can be found in Tables of The Paved Way, London, British Library, MS Or. 2806, fol. 33v. 48 Mathematical elements (mainly calculations) can be found in his The Wedge of Gold, an exegetical work, the conversion algorithm between Jewish and Muslim months mentioned above and the various astronomical tables, but in none of these texts is mathematics the main theme, it is only a tool for calculation. In the past, a mathematical treatise by the name A Procedure of Calculation (  ) was erroneously attributed to Isaac. See Lévy, Tony, ‘L’algèbre arabe dans les textes hébraïques (I), Un Ouvrage Inédit D’Isaac ben Solomon al     Arabic Sciences and Philosophy, 13, (2003), p. 301. 49 Isaac ben Solomon Ibn al- Leshon ha-Zahav, (A Wedge of Gold), London, British Library, MS Or. 10660, fol. 113v (IMHM 07975). The Hebrew edition of the text with commentary was published by Yaaqov Spiegel, Leshon ha-Zahav le-R. Isaac ben Solomon al- B.D.D., Volume 12 (Bar Ilan University), Winter 2000, pp. 5–34. I wish to thank Jesús del Prado Plumed for this reference.

14

THE EPISTLE OF THE NUMBER

type of mathematics used here involves basic arithmetic, mainly multiplication. 50 The types of numbers present in the text are integers and fractions. 51 The numeration system is either rhetorical or alphanumerical, i.e. according to the ABJAD system, in which the numerical value of a letter (1, 2, … 9, 10, 20, … 90, 100, …) corresponds to its position in the alphabet. Astronomy The study of astronomy was legitimized and widely encouraged in medieval Jewish circles because of its importance for the reckoning of the Jewish calendar. Indeed, one finds original Jewish astronomical writing in the Middle Ages. It is not surprising that the majority of the known copies of Isaac’s manuscripts belong to this domain. Isaac composed astronomical tables and wrote about astronomical instruments such as the astrolabe, the equatorium and the quadrant as well as some of his inventions or adaptations of existing tools. The Paved Way (   ). 52 This tract was composed in Syracuse in 1396 and about 25 copies of it survived, an impressive number for a medieval Hebrew text on science, indicating its status as a medieval “best-seller”. The Paved Way elaborates on the motion of the luminaries and it includes a set of user friendly tables on conjunctions and oppositions 53 as well as solar and lunar eclipses. This work was part of a rich astronomical tradition in the Iberian Peninsula and Provence in Isaac’s time, which consists of works by Immanuel Bonfils of Tarascon, Abraham Zacut and Judah ben Verga. This tradition derives from the Ptolemaic astronomy, elaborated in al-Andalus, then diffused to Jewish and Christian communities in Europe. This tradition gave birth to a separate branch of tables, which is independent of the Toledan or Parisian Alfonsine Tables. 54 A Precious Instrument ( ). Divided into two sections, consisting of four and thirteen chapters respectively, we find records of the construction and use of the equatorium, an astronomical instrument which was devised by Isaac in Syracuse in 1396 for the determination of the position of the planets with greater precision. It only requires the turning of dials, instead of cumbersome calculations with 59F

60F

To designate multiplication, Isaac uses only the term , whereas  , a calque of the Arabic , does appear in The Epistle of the Number. This may indicate that this exegetical work had been composed before Isaac came in contact with Arabic mathematics. 51 Geometric terms include  (circumference),   (thickness), * (area),   (width),  (length). The arithmetical terms used are identical to those in Book I of The Epistle of the Number. These terms are:   (approximately), +  (addition) and the fractions   (one fifth) and  (one sixth). 52 The title derives from a biblical expression [Proverbs 15:19]. 53 See the section on Jacob above. 54 José Chabás and Bernard R. Goldstein, Flavius Mithridates, 147–148, 170 (n. 4). 50

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astronomical tables, which are prone to arithmetical error by the (human) calculator. 55 In the colophon we read: Epistles on the precious instrument, composed by the wise man, Rabbi Isaac Ibn al-Adab, be his name in blessed memory, who says: the precious instrument is precious to you above all instruments of marvels. Its name is dear and pleasant. Isaac ben Solomon is like a father to you, and you are to him like a beloved child ( *). 56 62F

The Intermediate Instrument (  ). In this treatise, which includes 26 chapters, Isaac describes an instrument which is a combination of an astrolabe and the new quadrant invented by Jacob ben Makhir around the year 1300. 57 This new quadrant, which became very popular in Christian Europe, derives from the astrolabe by folding the stereographic projection of the heavens in half, twice. Isaac composed two epistles on the subject, one about the construction of the instrument and the other about its application. Isaac followed the doctrine of ar-}~~`   _ tables calculated by both al-  &   r-}~~`€ ‚ „ † ‡  ˆ‰ latitude 36°, even though Syracuse lies in latitude 37°. It is possible that latitude 36° was chosen for the convenience of calculation. 58 The Epistle of the Number (  ). In this text, Isaac seems to allude to Jacob ben Makhir. In the chapter on multiplication of numbers he comments on vertical multiplication on paper, where intermediate results cannot be erased (as is the case of wax boards), Isaac refers to the table representation of one type of vertical multiplication as The Owner of Wings ( ), 59 alluding to the title of Jacob ben Makhir’s important astronomical work Six Wings ( ). The numerical example given in folio 11r in The Epistle of the Number is the multiplication of 255,225 by 879. With a slight stretch of imagination, the shape of the table could perhaps remind us of a bird spreading its wings: 63F

64F

55

Bernard R. Goldstein, Astronomy in the medieval Jewish Spanish community, Between Demonstration and Imagination: Essays in the History of Science and Philosophy presented to John D. North, ed. L. Nauta and A. Vanderjagt. (Leiden, 1999), pp. 233–234. 56 A Precious Instrument, London, British Library, Or. 2806, fol. 11r (IMHM 06385).*   literally means ‘a plant of delight’, in the sense of ‘a beloved child’. See Goldstein, Bernard R., ‘Descriptions of Astronomical Instruments in Hebrew’, From deferent to equant: a volume of studies in the history of science in the ancient and medieval Near East in honor of E.S. Kennedy, ed. David A King and George Saliba, (New York, 1987), pp. 124–128. 57 See Goldstein, ‘Instruments’, pp. 121–123. 58 Spiegel, Haggadah shel pessach, pp. 16–17. 59 Lit. ‘a bird’.

2 6 2 1 0 5 8 7 9 3 2 7 4 5 2 3 5 4

7 9 9

5

8 7

2

3 8 4 3

2

8

2

2 4

2

THE EPISTLE OF THE NUMBER

2

16

The Jewish calendar A fragmentary text composed by Isaac includes a calendrical table of the Muslim months and their (various) equivalents in the Jewish calendar. 60 A preliminary study of this fragment shows that this text was composed around the year 1387, possibly  @ !  Q  -Faraj. In the margins we find references to the years 1430 and 1445, which fit the period when Al-Faraj was involved in the copying of Isaac’s texts. The aim of the devised algorithm is to enable one to find the Muslim month in which any 1 Tishri (the first day of the month in the fixed Jewish calendar) will fall. No algorithm can provide the exact date (i.e. the actual day in the Muslim month) because the Muslim calendar was and still is (for example in Saudi Arabia) based on lunar sighting. However, the lunar component both in the Muslim and Jewish calendars ensures that the algorithm will yield pretty accurate results. My initial study of this fragment indicates that Isaac provided a mathematization of conversion tables already found in Isaac Israeli’s Yesod Olam (The Foundation if the World). The latter is a lengthy calendrical scientific treatise composed in Toledo in 1310, which discusses astronomical and mathematical issues at length – laying solid foundations for understanding all aspects of the Jewish calendar. It is important to note that Yesod Olam was dedicated to no other than the RoSH, and as we know, Isaac was taught astronomy by the RoSH’s great-grandson Judah ben Asher the Second, and he may have even been part of the RoSH’s family, as mentioned earlier in the chapter. Regardless of the exact nature of the connection between Isaac and the RoSH, it is highly probable that Isaac had direct access to Israeli’s treatise. In fact, we find calendrical notes by

60

Raanan, Poems, p. 17. This algorithm has survived in one copy only as far as we know, see Rome, Biblioteca Casanatense, MS Ebr. 3082, folios 28v (table) and 43r (instructions and examples), with unrelated astronomical materials inserted in between. As far as I know, this calendrical fragment has never been published and I intend to publish it in the future.

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Isaac Ibn al-Adab which follow Yesod Olam in MS Poc. 368 in the Bodleian Library. 61 ‘A table to [find] the unknown molad’ 62 (  ). The molad (literally ‘birth’) refers to the new moon. The calculation of the molad, in particular the molad of the month of Tishri, the Jewish New Year, is of extreme importance in the Jewish tradition. The determination of the time of the molad is based on the calculation of the length of all lunations which elapsed from Creation until the desired molad. The value of an average lunation used is 29 days 12 hours and 793 parts of 1080 parts of the hour. Isaac devises methods that render the calculation of the molad easier. 63 69F

Commentaries and exegeses A discourse on the definition of the sayings (    ): in this tract Isaac writes about the meaning of the mind, the soul, angels and stars. 64 A comment on the seven precepts for the sons of Noah (  

-  ): these seven precepts (six negative and one positive) are perceived by Jews as the basis of the ethical code for non-Jews. 65 A commentary on the Azharot 66 composed by Solomon Ibn Gabirol. In it, Isaac declares that he “has only come to comment on the purity of the [author’s] language and the beauty of its subject”. 67 Exegesis of the Passover Hagaddah (   ). This book includes rules for the Seder dinner (  ), followed by an ample commentary on the Hagaddah. In the introduction, Isaac says that the purpose of his writing is “to explain some of the terms in the Passover Hagaddah, as requested by some of his friends”. As mentioned above, in the marginalia we find Isaac’s reference to the 70F

71F

61

This is most probably a Sephardi manuscript from the fifteenth century. The exact connection between the writing of the two Isaacs needs to be carefully investigated. One has to wonder whether the calendrical notes may even be in al-Adab’s own handwriting. See also the next footnote. 62 Oxford, Bodleian Library, MS Poc. 368, fols. 218r–219r (IMHM 19329). 63 1 Tishri may need to be postponed by a day or two from when the molad of Tishri falls, as a result of the religious prohibition that 1 Tishri fall on a Sunday, Wednesday or a Friday. Also, if the molad of Tishri occurs at noon or later (old molad) then 1 Tishri needs to be postponed to the following day and possibly postponed by another day if the previous restriction applies. 64 Spiegel, Haggadah shel pessach, p. 18. 65 Raanan, Poems, p. 18. 66 Azharot are didactic liturgical poems which treat the 613 precepts of the Jewish Law, 365 negative ones and 248, the number of parts in the human body ( ---), positive ones. In the Middle Ages it was common to dedicate a poem to one precept or precepts which belong to the same subject. The famous Jewish poet Solomon Ibn Gabirol wrote Azharot. 67 Spiegel, Haggadah shel pessach, pp. 18–19.

18

THE EPISTLE OF THE NUMBER

Reconquista (    ) sans plus. His language is clear and witty. He elaborates on every deed of the Passover Eve. This exegesis was probably written in Sicily, since the local habit to have a lengthy Kiddush is mentioned. 68 Isaac also discusses the Sicilian habit of adding dates, grains and seedless raisins, apples, almonds, hazelnuts and spices to the Passover horseradish (), instead of the habitual vinegar, in order “to render the Matzah tasty”. 69 In this tract, Isaac focuses on the interpretation of terms in the Hagaddah and unlike other exegeses of this kind, he does not collect previous commentaries; Isaac’s writing is characterized by a strong personal and independent interpretative style. 70 Only one copy of this exegesis is known to have survived and it dates from the end of the fifteenth century, now in a private collection. 71 Originally, this copy belonged to descendants of Isaac, as can be read from the signature of the owner, 'abib Ibn Isaac, and the signatures of other family members on the cover. There are very few Passover exegeses which survived the persecutions on the Iberian Peninsula. This fact enhances even more the value of Isaac’s exegesis and the unicum that survived. 75F

76F

7F

Poetry Isaac belongs to the generation of Jewish poets which seal the period of Sephardi Hebrew poetry and fight for its survival. He laments the decline of Hebrew poetry. In some of his poems, he expresses admiration to the early great poets such as Moses Ibn Ezra, Solomon Ibn Gabirol and Judah ha-Levi. He emphasizes the importance of poetry but insists that a poem must be novel and not a repetition of old ideas and writing styles. Unlike the earlier poets in al-Andalus (Muslim Spain), whose poems were highly rhetorical, full of puns and metaphors, poets in Christian Spain wrote in a didactic manner, conveying scientific, moral and social messages. 72 This literary trend comes as no surprise given that this was the prevalent poetical style in the Christian Castilian milieu, which clearly influenced Jewish poets. There are about ninety known poems written by Isaac, all of which were studied and categorized. 73 These poems have various natures: most of them are of didactic-ethical or religious. A few are humorous-satirical. In the poems dedicated to his sons on their wedding days, Isaac includes Maimonides’ Thirteen articles of the Creed and guides his sons to adhere to religious piety. He encourages his son Abraham to learn astronomy, study the rules of the Jewish calendar, which include the 68

Ibid, p. 23. Ibid, p. 29. 70 Ibid, p. 23. 71 At the private collection of the aDMoR from Karlin-Stolin in Jerusalem. 72 On this influence and the comparison between Jews poetry in Muslim and Christian Spain see Doron, Aviva, ‘Ha-Shir Bokhe ‘al mapalato, Itshaq al- – meshorer wivri beshalhey tif’eret sefarad’ (The poem laments its defeat – Isaac al- – a Hebrew poet at the end of the Glory of Sefarad) Leot Zikkaron, in Memorial volume dedicated to the memory of Aharon Mirsky, (Jerusalem 2006), pp. 333–345 (in Hebrew). 73 Raanan, Poems. 69

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calculation of the molad (new moon), ibbur (intercalation) and tequfot (i.e. seasonal turning points: spring equinox, summer solstice, autumn equinox and winter solstice), but he warns his sons to stay away from foreign sciences which are not in unison with Jewish sources. 74 Another poem concerns the thirty-two hermeneutic principles interpreting the Torah according to the Jewish tradition. Isaac also wrote poems as prefaces to books, proverbs, polemics and rhymed prose. He is believed to have written a humorous prayer in which all words start with the Hebrew letter nun . 75 No trace of this poem exists, but it is known that this type of poetry was common in the Iberian Jewish milieu, mainly as a brain teaser, a sort of medieval Sudoku. In one poem, Isaac complains about the religious interdiction to kill fleas on Sabbath. Two poems are addressed to the poet Zaraia Ibn Danon, as part of a polemic exchange between the two poets regarding who was a better poet, as mentioned above. Another category of poems includes strophic poems, i.e. poems in which the same rhyme scheme is repeated from one stanza to the other, such as in Poem on Esther and A diligent man. The latter is a well-known poem written with much wit, sarcasm and humour. In its composition, Isaac was probably influenced by Immanuel the Roman, who himself may have been influenced by the serventese and vanti poems, filled with bragging and vanity, a style which was common in Christian Italy but not in Christian Castile. In this poem, Isaac joyfully enumerates tens of social skills or trades which he himself practised. There are thirty-six stanzas, the first twenty-two of which open with the letters of the Hebrew alphabet in sequence and the last stanzas form the acronym +

  (be strong and of good courage). The narrator tells us about his frequent change of professions. This long list of occupations, which ranges from medicine to wafer making, can shed light on urban life styles among the Jews at that time. We also learn about the Jewish perceptions of those professions, such as in the case of doctors, who are characterized as cupid and womanizers. 76 Furthermore, Isaac is also attributed with a book on the art of poetry, of which all traces are lost. 77 81F

82F

83F

Medicine Isaac’s name was erroneously linked with the illustrious Arab medical writer Ibn !&X=– c. 1037), known under the Latin name Avicenna, since one of Isaac’s 

     \ ^     \   above. 78

74

Ibid, pp. 334, 337–339, 341–343. Raanan, Poems, p. 17. 76 Gutwirth, Eleazar, ‘Widows’, pp. 154–156. 77 Raanan, Poems, p.19. 78 Nevertheless, in Book I in the Epistle, Isaac compares ratios and numbers with temperaments, p‰| ˆ ‡ ‰  Š  Canon of Medicine. See footnote 26 above. 75

20

THE EPISTLE OF THE NUMBER

ISAAC’S LINGUISTIC SKILLS AS MANIFEST IN THE EPISTLE Isaac’s language in the Epistle is generally very clear and precise. Reading through the Epistle demonstrates Isaac’s great value as a pedagogue, since he often expresses his wish to be understood by his students and he finds alternative explanations and numerical examples. Whether the students were solely readers (including his friends who asked him to compose a mathematical treatise upon his arrival in Syracuse), or whether he was teaching these materials in groups, is not clear. Isaac’s sensitivity to language and capability to play with the language and create puns are manifest in his explanations of the etymology of several mathematical terms. For example, when explaining the meaning of prime numbers, he says that “They are also called deaf numbers, since they do not listen to the voice of divisors.” 79 Isaac’s central linguistic contribution to the Hebrew Epistle is the probable creation of novel mathematical vocabulary, mainly in the field of algebra. Given the phonological and lexical proximity between Arabic and Hebrew, Isaac often coined a Hebrew term that sounded close to its Arabic origin, while preserving the process of semantic extension. For example, in coining the algebraic term , opposition 80 Isaac was phonetically and semantically inspired by .  Regarding Isaac’s command of languages other than Hebrew, the impeccable translation of Talkh A  -  into Hebrew demonstrates his proficiency of the Arabic language. We also find textual proof that Isaac possessed at least partial knowledge of some Romance languages (spoken both in his native Castile and in Aragonese Sicily), perhaps he even knew Latin. Towards the end of folio 16v Isaac says that some languages allow for the form ‘one twentieth’, i.e. the name of a fraction with denominator greater than ten, which directly derives from the name of the integer in the denominator (this is the case of English). Neither Arabic nor Hebrew allow for a fraction corresponding to a number greater than ten, X, to be expressed as Xth, only as one [part] over X. The Romance languages, on the other hand, do permit to construct a fraction in one word which directly derives from its corresponding integer, in the way English does. Given Isaac’s geographical whereabouts, it is probable that by referring to “other languages” he meant either Latin, Medieval Castilian, Catalan or a certain Sicilian dialect. One twentieth in Latin is vicesimus, which derives from viginti, vigèsim in Catalan and vigesimo in Castilian. 81

THE ECHOES OF ISAAC’S WORK In some manuscripts, we find names of people and places relating to Isaac’s work: students, copyists and readers. The leading figures are described below:

79

The Epistle, fol. 2r. The Hebrew expression creates a pun and means that the prime numbers do not “obey” their divisors. 80 See lexicon. 81 Also see Isaac’s remark in fol. 17v.

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21

Q    -Faraj, mentioned above, is known to have been Isaac’s student. He copied a number of Isaac’s manuscripts in astronomy, philosophy, exegesis and poetry, leaving notes such as: I, the youngest of his students, Faraj ben al-Faraj, am writing… 82

We have an interesting family connection here, which seems to have contributed to the distribution of Isaac’s astronomical tables into Christian Italy: Nissim’s son, Samuel, a rather shadowy figure, was born in Caltabellota, Sicily. He converted to Christianity in the 1460s, changed his name to William Raymond of Moncada, the name of his first patron, the Count of Adrano, but then became known as Flavius Mithridates. He was a student at the University of Naples in 1473 and four years later, he arrived in Rome and met the Duke of Urbino, to whom he dedicated some astronomical tables in Latin. These tables derived almost entirely from Isaac’s The Paved Way ‰+ | ‚ ˆ‚ ‹`   +  ‰  Œ&‘ but Mithridates never mentioned the source of his tables. Around 1486, Mithridates translated kabbalistic tracts into Latin for Giovanni Pico della Mirandola, for whom he served as advisor. A rather controversial personality, he nevertheless evoked admiration thanks to his apparent genius and linguistic proficiency in Hebrew, Aramaic, Arabic and Syriac. 83 Muscato Bar Menaem completed copying An Epistle on A Precious Instrument ( ) in 1482. 84 Abraham Zacut (fifteenth century) refers to Isaac’s The Paved Way in his astronomical treatise The Almagest (   ). 85 Abraham ben 'ayim Gascon wrote a commentary on Isaac’s astronomical tables in 1542. 86 91F

82

A Table of the Muslim calendar and its analogue in the Jewish Calendar, Rome, Biblioteca Casanatense, MS Ebr. 3082, fol. 43r (IMHM 00072), probably composed in or around 1428. 83 Chabás, José and Goldstein, Bernard R. ‘Isaac Ibn al-'   ’„_ “‚s: The Diffusion of an Iberian Astronomical Tradition in the Late Middle Ages’, Journal for the History of Astronomy, xxxvii, (2006), pp. 147–148, 169 and Scandaliato, Angela, ‘Le radici familiari culturali di Guglielmo Raimondo Moncada, ebreo convertito del rinascimento, nella Sicilia del sec.  Una manna buona per Mantova, Man Tov le-Man Tovah, Studi in onore di Vittore Colorni, ed. Perani Mauro (Florence, 2004), 204–205. 84 A Precious Instrument, Paris, Bibliothèque Nationale, MS Héb. 1051, fol. 138v (IMHM 14656). 85 See, for example, Berthold Cohn, Der Almanach perpetuum des Abraham Zacuto (Strassburg 1918: Schriften der Wissenschaftlichen Gesellschaft in Strassburg, 32. Heft): p. 12 (German translation of Zacut’s introduction) and p. 46 (Hebrew introduction to Zacut’s treatise). Zacut was very well informed of the astronomical contribution by his Jewish predecessors, many of whom he mentions. For details, see José Chabás and Bernard R. Goldstein, Astronomy in the Iberian Peninsula: Abraham Zacut and the Transition from Manuscript to Print. Transactions of the American Philosophical Society, New Series, Vol. 90, No. 2. (2000), p. 49.

22

THE EPISTLE OF THE NUMBER

Rabbi Mordekhai ben 'ayim completed a copy of An Epistle on A Precious Instrument in 1422 i.e. during Isaac’s lifetime. 87 Rabbi Samuel bar Yoav from Modena wrote a commentary on The Paved Way. 88 Joseph ben Suleyman copied Isaac’s poem to his son Abraham in Baghdad in 1680. 89 Isaac’s astronomical tables and A Precious Instrument were used in Naples in 1492, 90 as well as in Baghdad, Jerusalem, Damascus and Palestine in 1738. 91 Mordekhai Finzi, the fifteenth-century Hebrew scholar, mentions Isaac’s astronomical works. 92 However, it is not known whether Finzi had read The Epistle of the Number or any of Isaac’s other scientific works.

SUMMARY Isaac lived during a tumultuous time for the Jewish people; yet, it is far from clear whether he was directly affected by the events, in particular the persecutions of 1391. From Castile to Sicily, passing through Muslim land and the Mediterranean Sea, Isaac transmitted science and poetry to the Sicilian Jewish world. His mathematical training in Muslim lands enabled him to present a rich tract on arithmetic and algebra to the Jewish community in Syracuse which derived from Ibn al-’s Talkh A ml al-isb. At least some of the members of this community apparently expressed interest in mathematics. Isaac was an astronomer, a poet, an exegete, a mathematician, a ‘calendar man’ and a coiner of mathematical terminology. The Wedge of Gold presents mathematical calculations within a religious context (biblical exegesis) whereas The Epistle of the Number exposes Arabic arithmetic and algebra in the Hebrew language outside a

86

Examples following the Tables in The Paved Way, New-York, Jewish Theological Seminary, MS 2571, fol. 1r (IMHM 28824). See also, Bernard R. Goldstein, ‘The Hebrew Astronomical Tradition: New Sources’, Isis 72. (1981), 240, 244 (reprint: Theory and Observation in Ancient and Medieval Astronomy, A collection of 24 essays, revised with a new preface. London: Variorum, 1985). 87 Paris, Bibliothèque Nationale, MS héb. 1065, fol. 70r (IMHM 31301). 88 Commentary on The Paved Way, Budapest, Magyar Tudomanyos Akademia, Kaufmann Coll. MS A13/14, fol. 242r (IMHM Fiche 029). 89 A Poem to the Wedding day of Abraham, Jerusalem, National Library of Israel, MS Sassoon 778, fol. 105r (IMHM 09556). 90 A Precious Instrument, Naples, Biblioteca Nazionale Vittorio Emanuele III, MS F 12, fols. 171r–172v (IMHM 11526). 91 Astronomical and Astrological Tables, Jerusalem, National Library of Israel, MS Sassoon 52, fol. 139r, (IMHM 08941). 92 Y. Tzvi Langermann, ‘The Scientific Writings of Mordekhai Finzi’, Italia, VII. Nr. 1– 2, (1988), 17, 20.

IBN AL-A'DAB

23

religious context. Isaac’s numerous astronomical works survived in abundance, in strong contrast to the paucity of traces of his purely mathematical writing. 93

93

This fact may corroborate Freudenthal’s conjecture concerning the marginality of mathematics among medieval Jewish communities. Freudenthal claims that Jews in twelfthcentury Provence could not have a-priori been interested in algebra because it was too foreign a science and irrelevant to religious practices. See ‘Science in the Medieval Jewish Culture of Southern France’, History of Science, xxxiii (1995), 36-37. The existence of The Epistle of the Number with very weak traces of algebra preceding it does not necessarily contradict Freudenthal’s general line of argumentation but in general, one should avoid argumentum ex silentio. For the discussion of previous traces of algebra in Hebrew see chapter 3, pp. 88–93.

CHAPTER 2: THE HEBREW AND ARABIC EPISTLES THE ARABIC SOURCE Identification of the Arabic source of The Epistle of the Number was complicated by the fact that neither author nor source title are explicitly evoked in the text. However, in the exordium to the Hebrew Epistle, Isaac recounts the story of a certain epistle in Arabic, which he later translated into Hebrew for the Jews of Syracuse. The only clue in the Hebrew text to its Arabic source is a reference to the later commentary to this unnamed Epistle by the same Arab author, entitled The Unveiler of the Veil (  ), which translates into Raf al-%  in Arabic. The latter is the title of a known commentary by a thirteenth-century mathematician, Ibn al- 

 on one of his mathematical tracts, Talkh A m l al-is b. 1 This key detail enabled Tony Lévy to trace The Epistle of the Number back to its Arabic source, Talkh A m l al-is b. 2 10F

10F

Isaac ben Solomon ben addiq Ibn al-Adab the Spaniard says: A Muslim scholar was once asked by some of his friends to compose a short treatise for them which would encompass all matters of the Science of the Number in a concise manner. He complied with their request and composed a very short epistle. He was doing marvels 3 in presenting its methods and in abbreviating it according to its matters and sent it to them. When the Epistle reached his friends, it was not accessible to them and it seemed to have transcended their comprehension. 4 They asked him to explain it. Sensing the limitation of their intellect, the author also reacted with cunning, by composing for them a commentary of the Epistle, which was so unusual and profound, that only a logician and a person who understands the giving of the causes of things could understand it. He named it

1

The complete title of Ibn al- 

’s commentary is &"

-%   '%* m  alis , i.e. The Unveiling of the Veil on the Methods of the operations of Calculation. 2 Lévy, ‘L’algèbre Arabe’, p. 288. 3 The Hebrew expression    is found in Joel 2:20, where it is translated, for example, in the King James Bible version, as “he has done great things.” I translated this expression in my own words in order to suit the context. I usually do not follow any standard translation of the biblical expressions in the introduction to The Epistle of the Number, but rather, I chose what I saw fit in the context of our text. 4 Lit. ‘the eyes of their wisdom’.

25

26

THE EPISTLE OF THE NUMBER

The Unveiler of the Veil. 5 The readers replied by the same coin saying that its name is The one who returns the Veil. 6 In return, he also wrote to them the following: “I am committed to making the effort to expose the subject matters from the source of their fundaments. However, I am not obliged to try to make the wild beasts 7 understand.” Later on, the Epistle spread among the wise, 8 for whom it became an epistle of utmost beauty. Its nature became known amongst them and they wrote on it many commentaries of many types, all very long. When I reached their countries I was dwelling in the Tents of Kedar 9 and that Epistle reached me. I studied it with of one of their sages. I also saw its author’s commentary and the commentary of others until I reached its inner treasures and revealed all its mysteries. [fol. 1v]

Ibn al-

 %-l Abbas Amad Ibn Muammad Ibn Utman al-Π - 

 ! 15 The text presents the six canonical equations with their corresponding solution algorithms, while discussing the procedure of normalization of the equations. Furthermore, two forms of abbreviated notation for algebraic species are discussed. It is also explained that equations of the third degree or a higher one cannot be solved unless no numbers are present and the equation is thus reducible to a quadratic equation. The next theme is algebraic expressions, which are formed by the addition and subtraction of algebraic species. The addition and subtraction of dissimilar species is carried out by the particle of addition and and the particle of subtraction less, respectively, no real operation can take place. The multiplication of algebraic expressions is carried out with the help of tables, and it is followed by the division of algebraic expressions. Only the division by a monomial is permissible, and one must never divide a lower species by a higher one. The last three folios of The Epistle of the Number, truncated at the end of folio 38v, include a series of problems of practical nature, which are reduced to the solution of linear or quadratic equations: six problems which reduce to the solution of the six equations, tricks to overcome problems which involve the division of a lower species by a higher one, one charity calculation, two distance-time-velocity problems and five problems of a horse purchased by several associates, as well as this is the only Hebrew text known to us which discusses arithmetical operations upon irrational quantities. See Tony Lévy, ‘The Establishment of the Mathematical Bookshelf of the Medieval Hebrew Scholar: Translations and Translators’, Science in Context, 10 (1997), 431–451. 15 Mohamed ben Musa, The Algebra.

A MATHEMATICAL COMMENTARY

41

the introduction of a further problem, at which point the unicum of the Epistle is truncated.

A DETAILED ANALYSIS OF BOOK II Given the novelty of most elements within Book II on the medieval Hebrew mathematical bookshelf, I have deemed it fitting to interpret the contents of Book II at length. The first part of Book II concerns two arithmetical methods for finding unknown quantities (the rule of three and the double false position). The second part is dedicated to algebra: algebraic species (a root/a thing, a square/an estate, a cube), algebraic operations (restoration, opposition and equalization) and the six canonical equations. Further on, the reader finds algebraic expressions. 16 Isaac introduces two different abbreviations for the algebraic species. One also finds various rhetorical problems of charity distribution, distance-time-velocity and the common purchase of a horse by several associates. All these problems are solved by means of algebra. For the sake of user-friendliness, I have combined a verbal description of the contents as well as formulae in modern mathematical notation. The table below presents how the symbols and algebraic terminology should be interpreted here. 17 Symbol

The meaning of these symbols in the context of

T he Epistle of the Number

x

The root, the unknown

x2

The square, the estate, the estate

x3

The cube

x4

The square square

Symbol 16

The meaning of these symbols in the context of

A general polynomial cadre is absent in this text. See Lévy, ‘L’algèbre arabe’, p. 292. I have endeavoured to reflect the medieval mathematical mind-set in an accessible way to the modern reader without committing the sin of anachronism. To corroborate the validity of my notational choice, I shall say the following: unlike the classical case of using algebra to describe Euclid’s Elements criticized by Unguru, in our case, there is no risk of using algebra to disguise another field (such as geometry), since we are dealing with algebraic notation of algebraic materials. See Sabetai Unguru, ‘On the Need to Rewrite the History of Greek Mathematics’, Archive for History of Exact Sciences, 15 (1975), 67–114. Here, a priori, we would only be prone to anachronism, i.e. the danger being that a careless writing of modern algebraic notation may no longer be loyally reflecting the algebraic content of the medieval tract. This risk is overcome here by specifically stating what each symbol and word means in The Epistle of the Number and the awareness that none of these symbols appear in the Epistle. The historiographical debate concerning the usage of algebraic symbolism to analyse the contents of ancient texts is outside the scope of this book. 17

42

THE EPISTLE OF THE NUMBER

T he Epistle of the Number

x

5

The cube square, the square cube

x6

The cube cube

x7

The cube square square

x8

The cube cube square

x9

The cube cube cube

xa

A species with rank a

xBIG

The large solution

xsmall

The small solution

xunique

The unique solution

 /‫׌‬

For every / There is

An algebraic expression

An algebraic expression is created when two or more algebraic species are added, subtracted or multiplied

x An algebraic expression with one variable e.g.

x  x2  x3

x An algebraic expression with two variables, e.g.

x An algebraic expression in which the species are related, e.g. a root plus a square plus a cube (of the same root) x An algebraic expression in which the species are not related, e.g. a root plus a square (whereas the root is not the root of the square)

x  y2 x1 , x 2 , x3 , x4

Four given unknown numbers

x1 x2

The ratio between x1 and x 2

=

Equals

z

Does not equal

An equation

One algebraic term, containing one algebraic species or more, is equal to another algebraic term with one or more algebraic species

Symbol

The meaning of these symbols in the context of

A MATHEMATICAL COMMENTARY

T he Epistle of the Number

l ( x) c

A linear equation with the unknown x

'1 ' 2

The first and second errors in the method of scales

signum('1' 2 )

The value of the multiplication of the errors, either deficient (negative) or superfluous (positive)

signum('1' 2 )  0

One error is superfluous and the other is deficient

signum('1' 2 ) ! 0

Both errors are either superfluous or deficient

O ( A) o B

Applying the operation O on A yields B

DŸE

D implies E

>

Larger than


th century), Ibn al- 

 (13th century) and al-–  , (Damas 1972), p. 14: “these exponents increase in the same proportion to infinity”. Also, see Franz Woepcke, 4?@@ ! \^ Traité d’algèbre par Aboû Bekr Mohammad Ben Alhaçan Alkarkhi, (Paris, 1853); reprint (Hildesheim, Zurich, New York, 1982), p. 48 and Lévy, ‘L’algèbre arabe’, p. 292. 28 This part is absent from the Arabic source. This categorization of numbers and algebraic expressions stems from an unknown source. 29 This does not appear in the Talkh. 30 See Lévy, ‘L’algèbre Arabe’, p. 293.

52

THE EPISTLE OF THE NUMBER

ƒ

the cubes. Each stands is by itself and is connected to the other and is also called a connected term.” After examining the next category of additive terms, in which the algebraic species are indeed related to each other, i.e. the root is the root of the square, etc., there seems to be no other possible interpretation except for that Isaac must have referred to algebraic expression of the type a root plus a square (not of the root) plus a cube (not related to the previous square or root), as one could express today within the polynomial context 31 x  y 2  z 3 . Let us look at the following textual elements, which corroborate this assertion. First, there does not seem to be another interpretation of “the roots are not [derived] from the squares”. If the roots are not those of the given squares, then they must be of other squares. Furthermore, Isaac later describes a category of additive terms, in which roots are indeed related to the squares and vice versa! Then, one could ask: what else could be the difference between the two cases except for a conceptual difference between the addition of species which are related and those which are  # The third type of numbers is a dditive . This term refers to a number 1 1 which is added to another one and relates to it such as in 1 ; the 4

ƒ

4

quarter is one quarter of the one. The corresponding algebraic term refers to any algebraic species which is added to related one(s) such as x in x  x 2  x 3 , where the root is the root of the square. The fourth type of numbers is subtractive . This includes a number 1 1 which is subtracted from another one, such as 2 in 10-2 or in 1 . A 4

4

subtractive algebraic term is the subtracted term within algebraic species such as x in x 2  x . There is no reference here to whether the terms or the numbers are related or not. The four types are summarized in the table on the following page:

31

Polynomials did not exist in the Middle Ages, of course!

A MATHEMATICAL COMMENTARY

Integer, isolated

For numbers

For algebraic expressions (as appears in the text)

For algebraic expressions (in modern notation)

2

A root, a square,…

x, x 2 ,...

5 in 4+5

A root plus a square (the root is not of the square)

 , Connected  

 

1 in 4 1 1 4

A root plus a square (the root is the root of the square)

Subtractive

4 in 5-4

 

1 in 1 1 4 4

A root in: a square minus a root

Additive

53

The essence of the definition

x  y2

Unrelated to its corresponding species

x2  x

Related to its corresponding species

x in x2  x

A term in which subtraction is involved

After the introduction of the four types of algebraic terms, arithmetic operations (addition, subtraction, multiplication and division) are applied upon isolated, additive and subtractive terms to create algebraic expressions. Division by additive and subtractive terms is problematic, as will be discussed later on. At this point, Isaac only tells us that algebraic expressions are formed by combining roots, squares etc. in the same way numbers are formed by the combination of units, tens and hundreds. The combinations of the various species can equal each other, in which case we obtain equations. There are six canonical equations. 4. Four sufficient conditions for an equation to be solvable [fols. 27v–28r] The following four conditions are stated as necessary and sufficient in order for a given equation to be solved: I. The species in one term must be different than those in the other term. II. One term must include one species and the other term must include either one or two species. III. If roots and squares are given, then the roots must be the roots of the squares. 32 IV. The rank of the algebraic expression in both terms cannot exceed two, i.e. the species can only be numbers, roots or squares. 33 32

This phrase corroborates the conjecture that Isaac was aware of the existence of other algebraic expressions of the form x  y 2 .

54

THE EPISTLE OF THE NUMBER

Definition of opposition and equalization: [fol. 28] Opposition is the subtraction of each species from its own until no similar species are found in both terms of the equation. Equalization consists in the restoration of the appropriate species to the second term which did not undergo restoration, and hence, it maintains the equality between the given terms. The following examples show different types of equations. An example when one term is subtractive and the other term is additive 4 x 2  2 x 3x 2  2 x restoration (4 x 2  2 x) o 4 x 2  2 x  2 x

4x 2

equalization (3 x 2  2 x) o 3 x 2  4 x

o 4x 2

3x 2  4 x

opposition (4 x 2

o x2

3x 2  4 x) o 4 x 2  3x 2

3x 2  3x 2  4 x

4x

The following examples represent various categories of equations, reduced to a canonical solvable form. The last result is one of the six types of equations, whose solution is presented in the next chapter. Examples where the dissimilar terms are subtractive x 2  x 4x  4 restoration( x 2  x)

x2

equalization (4 x  4) 5 x  4 o x2

5x  4

restoration(5 x  4) 5 x equalization ( x 2 )

o x2  4

x2  4

5x

Isaac also mentions a shorter method according to which the subtractives are exchanged and the particle of subtraction is omitted, and the rest is done by opposition and equalization. For example,

33

Isaac emphasizes that a given equation can be solved by means of restoration and opposition if and only if it contains only numbers, roots and squares. Equations of the third degree cannot be solved. See Lévy, ‘L’algèbre Arabe’, 292–293.

A MATHEMATICAL COMMENTARY

55

2 x 2  2 x 3x 2  24

o 2 x 2  24

3x 2  2 x

equalization (3 x 2  2 x)

o x2  2x

x 2  2x

24

Handling subtractives of the same species In addition to the method described above, Isaac mentions a shorter way, in which one subtracts the subtractive from the subtractive in the terms which belong to the same species. The subtractive become additive, as described below. An example of similar species equal in number 2 x 2  3 x 32  3 x o 2 x 2  3 x  3 x

32  3 x  3 x o 2 x 2

32

By using the shorter method, one obtains the result directly. An example of similar species unequal in number 2 x 2  3x

48  7 x

restoration(2 x 2  3 x)

2 x 2  3x  3x

equalization(48  7 x)

48  7 x  3 x

restoration(48  7 x  3 x) o 2x 2  4x

48  7 x  3 x  7 x

48

By using the shorter method, one obtains: 2 x 2  3x

48  7 x o 7 x  3x

4x o 2x 2  4x

48

5. The Six Canonical Equations 34 [fols. 28v–29r] In the next part, Isaac presents the six canonical equations, in which combinations of numbers, roots and squares are set equal. The six equations are: i. (Simple) Squares equal Numbers ax 2 bx ii. (Simple) Squares equal numbers ax 2 c iii. (Simple) Things equal numbers bx c iv. (Composite) Squares plus things equal numbers ax 2  bx 34

c

Al-˜‚†Œ`& † ‚ ˆ ‰ +  ‚ • ~_‰   ‡  ˆ‰` (rhetorically), as presented here. See Mohamed ben Musa, The Algebra, (London, 1831), pp. 6–13.

56

THE EPISTLE OF THE NUMBER

v. (Composite) Squares plus numbers equal things ax 2  c bx vi. (Composite) Things plus numbers equal an estate bx  c x 2 6. The normalization of quadratic equations Whenever the number of squares in the equation is not equal to one, Isaac explains how to convert it into 1 35 by dividing all coefficients by the coefficient of the square. 36 Chapter II: The procedure for the six equations The second chapter in Book II focusses on the algorithms which solve the six canonical equations. Isaac chooses to begin this chapter with a short discussion of the abbreviation of algebraic species, which he refers to as a ‘sign’ . This notation is not present in Talkh A m l al-isb. 37 152F

1. Abbreviations for algebraic species [fol. 29v] Isaac elaborates on how to abbreviate the notation of roots, squares and cubes, by using the first letter in the name of the species above its number, also expressed in Hebrew letters. The sign of roots is the letter  (first letter of roots, ), in English we would write R (Roots). The sign of estates is E ( for   ) and C is 

R

the sign of cubes ( for ). For example,  (= 2 ) designates two roots. A second notation appears four folios later [fol. 33]. Similarly to the first one, it sets the number of species below, but above, instead of the first letter of the species, its degree appears, all in Hebrew letters. Here, too, Isaac designates this notation by 

1

a ‘sign’ (). So, for the example given above of two roots we will have  (= 2 ). The second notation also identifies the corresponding name in a unique manner, just as the first one does. Unlike the first notation, however, the second one does not depend on the actual name of the species in a given language, but it is based upon the letter representing the degree which corresponds to the species. This notation is more abstract than the first one because it involves the actual degree of a species, and not its name. It represents a small step towards symbolic notation. It is important to emphasize that none of these abbreviations (or any other abbreviations) are known anywhere else within the medieval Hebrew mathematical

35

What we would call today to normalize the equation. For example, for the fourth type of equation, i.e. the first of the composite one: b c :a 2 o ax  bx c, a z 0, 1  x2  x a a 37 This is found neither in the Talkh nor in &"

-% €

ˆ  ˜‚% describes Ibn al- 

 ˆ_ ‰ _  | ‰‰ ˆ‰ ‡ + ‰ ‰   _ ‚ Isaac borrowed this from other sources, but they are unknown to us. See Lévy, ‘L’algèbre arabe’, pp. 295–296. 36

A MATHEMATICAL COMMENTARY

57

corpus. One also notes that except for some examples he gives for both abbreviations, Isaac does not make use of these notations. 2. Solving the simple cases (Types 1–3) [fols. 29v–30r] The procedure to solve the simple equations is to divide each coefficient of the equation by the number of squares, if present, or by the number of roots, if there are no squares. By doing so, one obtains the root in the first and third cases, and the square in the second case. ax 2

bx o x

b a

ax 2

c o x2

c a

bx

cox

c b

Examples First type: x2

6x o x

6 1

6 o x2

36

The verification is carried out by inserting of the solution into the original equation: 6 2 6 ˜ 6 36 14 2 x 2 14 x o x 7 o x 2 49 2 1 2 x 2

4x o x

Second type: x 2 25 o x

4 1 2

8 o x2

64

5

4x2

64 o x 2

64 16 o x 4

1 2 x 2

18 o x 2

36 o x

6

4

58

THE EPISTLE OF THE NUMBER

Third type: x 13 o x 2

169

20 o x

5 o x2

25

1 x 10 o x 2

20 o x 2

400

4x

The root and the square can be a fraction, too: 1 2 1 4x 2 2x o x o x2 4 2 4 3. Solving the fourth type of equation [fol. 30r] 2

x 2  bx

cox

b §b· ¨ ¸ c  2 2 © ¹

An example of the fourth type 10 x 2  10 x 56 o 5 o 52 2 o 81 9 o 9  5 4

Verification:

16  10 ˜ 4

25 o 25  56 81 x o x2

16

56

4. The normalization procedure: reducing the number of squares to one [fol. 30r] We learn that normalization is not necessary in the simple equations but is indispensable for the composite ones. 5. Solving the Sixth Type [fol. 30r] 2

bx  c

x2 o x

b §b· ¨ ¸ c  2 2 © ¹

The algorithm for the sixth type of equation is given right after the fourth one because they are similar.

A MATHEMATICAL COMMENTARY

59

Example of the sixth type

10 x  56

x2 o

o 81

10 2

5 o 52

25 o 25  56 81

9 o 9  5 14

Verification: 10 ˜ 14  56 14

2

x

196

6. Solving the fifth type [fol. 30v]

x 2  c bx 2

o x BIG

b §b·  ¨ ¸ c 2 ©2¹

o x small

b §b·  ¨ ¸ c 2 ©2¹

2

For this type of equation, there are two possible solutions. They collapse to a single solution whenever the square of half the number of roots equals the number. Example of the fifth type, whereas the reader can opt for either solution 6 x2  8 6x o 3 o 32 9 o 9  8 1 2 o 1 1 o 3 1 4

x 2  8 6x o

6 2

3 o 32

o 1 1 o 3 1

2

xBIG

9 o 98 1 xsmall

Special cases for type 5 Case 1 (both solutions collapse to one solution): 2 b §b· c Ÿ Unique solution : xunique ¨ ¸ 2 ©2¹ For example, 2

§b· x 2  16 8 x o c 16 ¨ ¸ Ÿ Unique solution : xunique ©2¹ Case 2 (there is no solution): 2

§b· c ! ¨ ¸ Ÿ No solution  ©2¹

4

60

THE EPISTLE OF THE NUMBER

For example, 2

x 2  20 8 x o c

20 ! 16

§b· ¨ ¸ Ÿ No solution ©2¹

A Shorter method for the fourth type (no normalization is required) 38 [fol. 31r] After previously stating that normalization is obligatory for the composite types but not for the simple cases, Isaac now presents a method by which normalization is not necessary for any type of equation. 2

x

b §b· ca  ¨ ¸  2 2 © ¹ a

Examples: 2

x 2  10 x 56 o x

§ 10 · 10 56 ˜1  ¨ ¸  2 ©2¹ 1

4 o x2

16

2 o x2

4

2

3 x 2  6 x 24 o x

6 §6· 24 ˜ 3  ¨ ¸  2 2 © ¹ 3 2

1 2 x  4x 2

24 ˜ 24 o x

1 §4· 4 ¨ ¸  2 ©2¹ 2 1 2

4 o x2

16

A Shorter Method for the Fifth Type (without Normalization) [fol. 31r] 2

xBIG

38

b §b·  ¨ ¸  ac 2 ©2¹ a

Ibn al- 

  ‚ ‰`Œ‰  ‰‡‰| ˆ‰  • ~_‰ € ¡‰†„ Isaac presents here a solution procedure for the composite equations, in which no normalization is necessary. He says that he has adopted this method from one Muslim scholar,  , see fol. 31r.

A MATHEMATICAL COMMENTARY

61

2

b §b·  ¨ ¸  ac 2 ©2¹ a

x small

Examples: 3 x 2  9 12 x o

o 9

12 2

36 o 9 ˜ 3

27 o 36  27

9

3 o 6  3 9 o 9 : 3 3 xBIG

3 x 2  9 12 x o

o 9

6 o 62

12 2

6 o 62

36 o 9 ˜ 3

27 o 36  27

9

3 o 6  3 3 o 3 : 3 1 xsmall

The verification is done as follows: 3 ˜ 32  9 12 ˜ 3 36 3 ˜ 12  9 12 ˜ 1 12 12 3 x 2  9 12 x o 6 o 6 2 36 o 9 ˜ 3 27 o 36  27 9 2 o 9 3 o 6  3 9 o 9 : 3 3 xBIG 1 2 x  16 2

6x o

6 2

3 o 32

4 1 2

o 1 1 o 3 1 4 o

1 2 6 x  16 6 x o 2 2

3 o 32

6x o

6 2

3 o 32

o 1 1 o 3 1 4 o

1 2 x  16 2

6x o

6 2

4

1 2

8o98 1

1 2

8o98 1

x BIG

9 o 16 ˜

xsmall

9 o 8 ˜1 8 o 9  8 1 4 1

3 o 32

o 1 1 o 3 1 2 o

8

2 1 2

o 1 1 o 3 1 2 o

x2  8

9 o 16 ˜

4

xBIG

9 o 8 ˜1 8 o 9  8 1 2 1

2

xsmall

62

THE EPISTLE OF THE NUMBER

A Shorter Method for the Sixth Type (without Normalization) [fol. 31v] 2

x

b §b· ca  ¨ ¸  2 ©2¹ a

Example 1: 2

8 x  16 3 x 2 o x

8 §8· 16 ˜ 3  ¨ ¸  2 ©2¹ 3

4 o x2

16

Example 2: 2

2x  6

1 §2· 2 6˜ ¨ ¸  2 ©2¹ 2 1 2 x o 1 2 2

6 o x2

36

Example 3: 2

10 x  56

10 § 10 · 56 ˜1  ¨ ¸  2 2 © ¹ x2 o 1

14 o x 2

196

Chapter III: Algebraic expressions First, the reader is reminded that algebraic species can be added, multiplied, subtracted and divided. This way, algebraic expressions are formed. The last two operations, however, are subject to some limitations, as we shall see. 1. Addition of similar species [fol. 31v] Similar species are added as numbers. For example: two roots plus fifty roots are added in the same way as the numbers two and fifty are, and this results in fifty two roots, in modern notation: 2 x  50 x 52 x 2. The particle of addition and † [fols. 31v–32r] Isaac distinguishes between the addition of similar species, carried out the same way as with numbers, and the addition of dissimilar species. When dissimilar species are added, they cannot be added as integers, or reduced in any way and hence, they must be connected in a different manner. This is where the particle of addition and comes into play. I have designated this particle by † .. Case 1: addition of two dissimilar species by the particle of addition 100 x 2  75 x 100 x 2 † 75 x

†

A MATHEMATICAL COMMENTARY

63

Case 2: multiple terms where each term has a corresponding similar species (3 x 2  5 x)  (8 x 2  16 x) 11x 2 † 21x

Here the similar species are added as integers, i.e. squares with squares and roots with roots. Then, the dissimilar species are added by † . Case 3: all species in the addition are dissimilar (8 x 2  3 x)  (4 x 3  15) 8 x 2 † 3 x † 4 x 3 † 15

Here, only

†

is used to join all dissimilar terms.

Case 4: not all terms have a corresponding similar species (3 x 2  4 x)  (4 x 2  16)

7 x 2 † 4 x † 16

In this case we add the similar species as integers and join the result to the isolated dissimilar species by † . Case 5: subtractives are involved in the addition. For example, in 2 x 2  4 x , 4 x is the subtractive species and 2x 2 is the nonsubtractive species. Before introducing the particle of subtraction, Isaac tells us that for similar subtractives, one subtracts the smaller from the larger, whereas dissimilar subtractives do not undergo subtraction at all. For example, (3 x 2  7 x)  ( x 3  15 x) x 3 † 3 x 2  22 x Case 6: the algebraic expression involves only similar additives and subtractives The general rule given here is to add similar species as integers and then join dissimilar ones by the particle of addition and. Whenever a subtractive algebraic term is given, one must be aware that it is not possible to subtract a larger quantity from a smaller one. This rule is explicitly mentioned in Book I. 39 Thus, whenever subtractives are involved in an algebraic expression, one should first subtract the smaller similar subtractive from the larger one as integers. The rest of terms are left without subtraction. Isaac does not yet reveal that subtraction of dissimilar species will be done by the particle of subtraction less, so he simply tells his readers at this point that no subtraction takes place. For example, (3 x 2  6 x)  (7 x 2  5 x) 10 x 2  11x Case 7: The subtractive species in the one term is similar to the non-subtractive species in the other term In this case, one subtracts the smaller from the larger and joins the rest as necessary. Numerous examples are shown, each representing a variation in the addition of algebraic expression by the particle of addition.

39

Fol. 6v.

64

THE EPISTLE OF THE NUMBER

Examples: ( x 2  7 x)  5 x

x 2  2x

( x 2  7 x)  11x

x 2 † 4x

( x 2  4 x)  ( x 3  8 x)

x3 † x 2 † 4x

( x 2  8 x)  ( x 3  4 x)

x3 † x 2  4x

( x 2  4 x)  (5 x  15)

x 2 † x † 15

( x 2  4 x)  (5 x  15)

x 2 † x  15

( x 2  6 x)  (4 x  15)

x 2  2 x † 15

x 2  4 x  21

x 2 † 21  4 x

( x 2  6 x)  ( x 3  2 x 2 )

x 3 † 3x 2  6 x

( x 2  5 x )  ( x 3  2)

x 3 † x 2  (5 x  2)

When algebraic expressions with numerous terms are given one is to apply the procedure on two terms first and then continue on the rest two by two. Isaac meticulously covers a multitude of cases concerning the particle of addition through a wide range of examples. The particle of subtraction less 4 [fols. 32r-33r] The particle of subtraction less, designated here by 4 , is the mirror particle of † for subtraction, i.e. it is used to denote the subtraction of dissimilar species. Similar species are subtracted as integers, for example, 3 x 2  2 x 2 x 2 and larger quantities cannot be subtracted from smaller ones. We will see later, however, that for dissimilar species, it may be possible, under special conditions, to carry out the subtraction of a larger species from a smaller one. This is, of course, a-priori impossible in numbers. Various cases of subtraction are presented next. Case 1: The subtractive species is dissimilar to the non-subtractive species In this case no subtraction can take place and the particle of subtraction the impossible subtraction. Examples: 2 x 2  5x

4

marks

2 x 2 45 x

( x 3  4 x)  5 x 2

x 3 † 4 x45 x 2

( x 3  4 x)  (5 x  15)

x 3 † 4 x4(5 x † 15)

Case 2: Permissible values for algebraic expressions What follows is particularly interesting since it involves the reference to admissible values for a given algebraic expression which includes subtraction. First, Isaac

A MATHEMATICAL COMMENTARY

65

emphasizes that it is not possible to subtract a larger number from a smaller one and that the same holds for algebraic expressions. He examines the case x 3  (5 x 2  11x  15) , stating that the root cannot have an integer value smaller than 7. We could rewrite Isaac’s claim in modern notation as follows: x t 7, x  N o x 3  (5 x 2  11x  15) ! 0 . One must remember that the domain of numbers in this chapter includes positive integers and fractions. In his explanation, Isaac claims that in general a cube is greater than a square. This, of course, is true only for numbers greater than 1 i.e. x 3 ! x 2 x ! 1 . However, Isaac makes no mention of that. He further claims that in the algebraic expression x 3  (5 x 2  11x  15) the root must be greater than seven, without any indication how this result was obtained. Indeed, within the realm of integers, x 7 is the smallest integer such that the value of the given algebraic expression x 3  (5 x 2  11x  15) is positive. 40 Isaac presents no justification of the result. Also, the result presented by Isaac is probably based on numerical experimentation and it does not result from the application of a general method of finding permissible values of algebraic expressions, a notion that he does not mention explicitly. 41 Case 3: Subtraction between similar and dissimilar species When similar and dissimilar species are involved in the subtraction, one needs to subtract each species from its similar ones and connect the remaining dissimilar species by the particles of addition and subtraction. Examples: ( x 2  7 x)  (4 x  15) ( x 2  4 x)  (7 x  15)

x 2 † 3 x415 x 2 4(3 x † 15)

Case 4: Subtracting dissimilar subtractive and non-subtractive species If subtractive species appear in both terms or in one of them, then one should add the subtractives of each term to both terms simultaneously and then carry out the subtraction.

Indeed, x 6 yields a negative value -45 and the algebraic expression is monotonous. However, one may suspect that Isaac did not have these ideas in mind. Rather, this is an example of a permissible value for an algebraic expression, possibly obtained by numerical experimentation rather than within a solid theoretical framework. 41 See case 7, which will be even more intriguing. 40

66

THE EPISTLE OF THE NUMBER

Examples: (2 x 2  5 x)  ( x 2  3x)

2 x 2 † 8x  x 2

2 x 2  2 x  ( x 2  8 x)

2 x 2  ( x 2  10 x)

( x 3  21)  ( x 2  10 x)

x 3 † 10 x † 214x 2

( x 3  9 x)  (2 x 2  25)

x 2 † 8x x 2 410 x

x 3  (2 x 2  9 x  25)

x 3 42 x 2 49 x425

In the following example, both terms are subtractive: ( x 2  15)  (6 x  12)

( x 2  12)  (6 x  15)

x 2 4(6 x † 3)

Case 5: subtracting dissimilar subtractive species If both terms are subtractive, then the way to overcome this is by removing the subtractives in both terms and adding them to the other term. For example, ( x 3  24)  ( x 2  6 x)

( x 3  6 x)  ( x 2  24)

Case 6: subtracting a subtractive from a non-subtractive

2 x 2  ( x 2  7 x) x 2  (2 x  5)

x2 † 7x x 2 † 542 x

Case 7: subtraction of a higher species from a lower one Throughout the chapter, Isaac emphasizes that the subtractive term is of a lower species than the subtracted term, e.g. a thing can be subtracted from the corresponding square, but not the other way round. Here, however, he says that there are cases where it is indeed possible to subtract a higher species from a lower one. Isaac gives an example, which is somewhat mathematically incoherent, in part due to fact that the corresponding part in the manuscript is corrupt: Know that although it is right and customary that the subtractive in the problem be of smaller species rather than the larger one, such as a square minus roots, a square minus a number, roots minus a number and a cube minus all those species, if one still wants to ask about the number of roots minus a square or the number of estates minus a cube, he may do so. However, roots are always roots of the estate and estates are estates of the cube. However, the roots are larger

    \    ‚#„\ 42 and according to this 4 roots will remain. Even though one does not use the common custom in

The Hebrew says

   -   - , i.e. from 3 by a square minus 2 squares from that square, which seems like a corrupt copy. In what follows I suggest a possible reconstruction [ten roots minus a square, the root is 6]. 42

A MATHEMATICAL COMMENTARY

67

this case of reducing the problem to one of the 6 types of equations, the result is the same. [fol. 33r]

In this chapter, I propose a possible reconstruction of the corrupt text. My reconstruction is based on the context and it matches the legible numerical results in the unicum. ‚„†#‡  *\  ˆ\\‰ !Š

If this reconstruction is correct, then this means that for the expression 10 x  x 2 , x 6 yields 10 ˜ 6  6 2 24 4 ˜ 6 4 x . I.e. for x 6 one obtains an acceptable expression, i.e. an expression whose value is positive. In contrary to case 2, in which the entire domain of definition was given, here only suitable value is given, 6. The entire domain of definition consists of integers between 1 and 9. In case 2, Isaac presented the complete range of possible integers, whereas here only one permissible value of out several is given. This is quite perplexing. Since Isaac does not give a solid framework for type of calculation, the same questions as in case 2 arise here. The brevity of cases 2 and 7, contrasting with lengthy elaborations in other difficult issues elsewhere, corroborates the assumption that Isaac may have not been aware of the complexity of these problems, and that all he had in mind is to show examples permissible values. 43 At the end of this chapter, Isaac provides examples of how to reduce algebraic expressions which include addition and subtraction into one of the six solvable forms. If both terms contain a subtractive, then one transfers them to the other term of the equation by using opposition. For example from 6 x  x 2 20  4 x we obtain 10 x 20  x 2 and from 3 x 2  8 4 x 2  6 x we obtain x 2  8 6 x , the second type of the composite equations. Chapter IV: The multiplication of algebraic expressions 1. The degrees of species [fol. 33r] Species, just as integers, bear different ranks and to each rank corresponds a degree  . There is a one-to-one correspondence between a degree and its name. The degree of things (roots) is one, the degree of estates (squares) is two and that of cubes is three. Higher degrees are composed of the three basic ones. A general rule for their denomination is alluded to by the expression ‘and so on’. 43 The given expression is positive for any positive root smaller than 10. It is not clear why the value 6 was chosen and why Isaac did not give the whole domain of definition, i.e. all permissible integers for this algebraic expression. It is also not clear how he obtained this result, whether he experimented with various numbers until reaching a good value, or perhaps he copied this example from a source, which, unfortunately, I have not been able to identify.

68

THE EPISTLE OF THE NUMBER

2. The names of the degrees [fol. 33] To find the corresponding name of the species of a higher degree, Isaac distinguishes between four cases. Case 1: The degree is divisible by 2 but not by 3 Let us designate this type of degree by 2n, n  N such as 4 or 8. In this case one assigns to each 2 a square, resulting in a square multiplied by itself n times. Thus x 4 corresponds to a square square and x 8 corresponds to a square square square square etc. Case 2: The degree is divisible by 3 but not by 2 The degree is of the form 3n, n  N such as 9 or 15. In this case, one assigns to each 3 a cube, resulting in a cube multiplied by itself n times. Thus, x9 corresponds to a cube cube cube and x15 corresponds to a cube cube cube cube cube etc. Case 3: The degree is divisible by both 2 and 3 The degree is of the form 2m or 3n , m, n  N such as 12. In this case one can either assign to each 2 a square or 44 assign to each 3 a cube, resulting in a square multiplied by itself m times or a cube multiplied by itself n times. Thus, x12 is a square square square square square square or cube cube cube cube etc. Case 4: The degree is not divisible by either 2 or 3 The text discusses degrees which are not divisible by 2 or 3, such as 5, 7, 11. Degree 5 is a square cube or a cube square. x 7 is a square square cube or a cube square square. For degrees 7 and 11, Isaac recommends to start counting by squares, otherwise one is left with things, i.e. cube cube thing, a form he apparently does not recommend. 3. The multiplication of species by each other [fol. 33v] When multiplying one species by another, one needs to add the corresponding degrees. The rule is x a ˜ x b x a b For example: x1 ˜ x 2 x1 2 x 3 ; x1 ˜ x 3 x13 x 4 ; x 2 ˜ x 2 x 2 2 x 4 for a, b ! 0 integer. 4. The multiplication by a number [fol. 33v] The multiplication of a species by a number does not change the species of the multiplied term i.e. for any number b , b ˜ x a  Species( x a ), a t 1 . For example, 9 ˜ 2 x 2 18 x 2 . 5. Equations of third and fourth degrees which do not include numbers [fol. 33v] Isaac presents equations with cubes and square squares that do not include numbers. These equations can be reduced to one of the six quadratic equations. 45 The reduction is carried out by dividing each term in the equation by x a , whereas a 44

This is an exclusive ‘or’. This is of course possible if and only if the difference between the smallest and largest degrees of the equation does not exceed 2. 45

A MATHEMATICAL COMMENTARY

69

is the smallest degree present in the equation. Let us consider the following equation and the division of each of its elements by a square: :x 4 3 2 2 x 4 x  12 x o x 4 x  12 resulting in the fourth type of equation, which is solved by opposition and equalization. Another example: for x 2  x 3 12 x one reduces the least of degrees, 1, from all species, obtaining x  x 2 12 . 2

6. The multiplication of an additive by an additive, a subtractive by a subtractive and an additive by a subtractive (the rule of signs) [fol. 34] The multiplication of an additive or a subtractive by itself is additive, and the multiplication of an additive by a subtractive is subtractive, i.e.  ˜  ,  ˜  and  ˜  . 46 ( x  10) ˜ ( x  10)

x 2 4100

( x  10) ˜ ( x  10)

x 2 † 100420 x

7. Multiplication of algebraic expressions with tables [fol. 34v] Isaac presents a procedure for the multiplication of any two algebraic expressions through the use of tables. Two tables are drawn, each containing three rows. In the first and third row of the first table, one writes in one row the degrees of the multiplied algebraic expressions in ascending order. Then, one inserts the coefficients of the multiplied expression and the multiplier in their corresponding place according to their degree. In the first row of the second table, one writes again all degrees up to the highest one attainable in the multiplication. Then, one multiplies each coefficient in the multiplied with those of the multiplier and inserts the result in the corresponding column in the second table. For example, multiplying 2 squares by 3 roots yields 6 cubes, so 6 should appear under the column belonging to 3. Note that if any of the algebraic expressions includes a number, one writes it in a separate column with 0 at the top. 47 If both terms are subtractive, the answer is additive, but if one number is additive and the other is subtractive, the answer is subtractive and one notes it on an additional superior line in the second table, as shown in example 2 below. Example 1: 48 (3 x 3  7 x 2  10 x) ˜ (9 x 3  6 x 2  5 x)

27 x 6  18 x 5  15 x 4  63 x 5  42 x 4  35 x 3  90 x 4  60 x 3  50 x 2

(3 x 3  7 x 2  10 x) ˜ (9 x 3  6 x 2  5 x) 46

27 x 6  81x 5  147 x 4  95 x 3  50 x 2

Ibn al- 

 ‡„  ‰‡  +‰‰ˆ ‰ ‚ † | ‡ `     † algebraic term a rest, see Ahmad Djebbar, L’algèbre arabe, (Paris, 2005), pp. 90–91. 47 The appearance of the zero at the table of multiplication of algebraic species is extremely interesting and is elaborated on in the appendix. 48 Also, see Lévy, ‘L’algèbre Arabe’, 297–298.

70

THE EPISTLE OF THE NUMBER

The two tables shown in The Epistle of the Number for the above calculations are the following: 1

2

3

10

7

3

5

6

9

1

2

3

Cambridge University Library, Hebrew MS, Add.492.1, fol. 34v. Reproduced by permission of the Syndics of Cambridge University Library

0

1

2

3

4

5

6

50

35

15

18

27

60

42

63

90 50

95

147

81

27

Cambridge University Library, MS Heb. Add. 492.1, fol. 34v. Reproduced by permission of the Syndics of Cambridge University Library

A MATHEMATICAL COMMENTARY

Example 2: (2 x 2  3 x) ˜ (3 x 3  4 x 2 )

6 x 5  8 x 4  9 x 4  12 x 3 2

1

2

less

3

3

less

4

3

2

Cambridge University Library, MS Heb. Add. 492.1, fol. 34v. Reproduced by permission of the Syndics of Cambridge University Library 9 8 3

4

12 12

5 6

less 17

6

Cambridge University Library, MS Heb. Add. 492.1, fol. 34v. Reproduced by permission of the Syndics of Cambridge University Library

71

72

THE EPISTLE OF THE NUMBER

Example 3:

(5 x 3  3x 2 ) ˜ (4 x 2  2 x)

20 x 5  6 x 3  10 x 4  12 x 4 2

3

3

5

2

less

1

4 2

6

10

3

4

5

12

20

2

20

less 6

20 x 5  2 x 4  6 x 3

Chapter IV: Division 1. The division of a monomial by another monomial [fols. 34v-35r] Here the division of algebraic species is carried out by the subtraction of their corresponding degrees:

xa xb

x a b

Isaac emphasizes that the smallest of species which has a degree is the root, followed by the square, the cube and so on. Within our medieval context there is no conceptual connection between a number and the exponent zero. Isaac reminds the readers that there are two types of division: the first one is the division of a larger number by a smaller one, simple division, or the division of a smaller number by a larger one, i.e. denomination. Algebraic species can be divided, too, but one should not divide a lower species by a higher one. 2 For example, 10 x

5x

2x

The verification of the result is carried out by multiplying the result by the divisor and obtaining the divided term. The correspondence in the degrees is easily verifiable. 2. The division of monomials that have the same degrees [fol. 35r] When identical species are divided by one another, the result is a number. bx a cx a

b c

This rule is not incorporated into the general rule of division of a monomial by another monomial in the previous section. No connection is made between the two,

A MATHEMATICAL COMMENTARY

73

an this corroborates our understanding that numbers were conceived as separate entities, without any degree. For example: 10 x 5x

2

3. The division of a monomial by a number [fol. 35r] The division of a given species by a number does not change its degree: xa  species( x a ), a t 1 b

4. The division of a subtractive algebraic expression by another one, which contains only one term [fol. 35r] In this case, one should divide each species of the divided term separately and subtract the results i.e. a b c

a b  c c

Whereas a, b, c are algebraic expressions. For example: 4x 2  6x 2x

2x  3

5. The impossibility of dividing a lower degree by a higher one [fol. 35r] The text clearly states that the division of a lower degree by a lower degree by a higher one is not possible. This Leitmotif throughout the section of subtraction is now applied to division. In subtraction, one cannot a priori subtract a higher species from a smaller one 49 and similarly, a priori, one cannot divide lower species by higher ones. 6. Dividing by a subtractive [fol. 35r] Even though it is generally forbidden to divide by expressions which contain a subtractive, in a certain type of problems in this chapter, it is allowed.

49

Although a special case was presented earlier.

74

THE EPISTLE OF THE NUMBER

PART III: PROBLEMS OF PRACTICAL NATURE Chapter I: Theory and practice for the solution of practical problems 1. Problems illustrating the six equations [fols. 36v–38v] Isaac elaborates on a series of rhetorical problems: the partition of the number ten, charity donation, encounter (involving distance-time-velocity) and five problems of a horse purchase among associates. The unicum is truncated immediately after the fifth problem of horse purchase by the words “another example”, so there must have been originally a sixth problem copied into a new quire. The algebraic methods required to resolve these problems were already explained in the second book of the Epistle. 50 Problem #1 The first problem: ten, we partition it into two parts and we multiply one part by itself. Then we multiply one part by the other. The result of the multiplication of the one part by itself equals 4 times the result of the multiplication of the one part   !Œ    $=# [fol. 36v]

Solution: by denoting one part of ten by x and the other by 10  x then x2

4 x(10  x) o x 2

40 x  4 x 2 o 5 x 2

40 x o x 2

8x

This is the first type of equation. By using the known algorithm for this case we obtain the solution x 8 and 10  x 2 and indeed, 8 2 4 ˜ 8 ˜ (10  8) Problem #2 The second problem: ten, partition it into two parts. Multiply one of the parts by itself as well and the result of the first multiplication, which is the multiplication of the ten by itself, equals sixteen times the amount of the other multiplication. [fol. 36v]

Solution: by denoting one part of ten by x and the other by 10  x then 16 x 2 100 . This is the second type of equation. By using the procedure for this case, one obtains

50

Many of the problems and equations in the Epistle are extant in treatises written by al-˜‚†Œ`& -˜‘&   % ˜` _ ‚| † ‰ | ‚ ‰_ ˆ‰  since this type of problems was common in the Arabic mathematical tradition. See Warren Van Egmond, ‘Types and Traditions of Mathematical Problems: A challenge for Historians of Mathematics’, Mathematische Probleme im Mittelalter (Wiesbaden, 1996), 400. See also Lévy, L’algèbre arabe, p. 297.

A MATHEMATICAL COMMENTARY

x2

75

1 1 1 1 1 6 o x 2 o 10  x 7 o 16 ˜ (2 ) 2 16 ˜ 6 100 4 2 2 2 4

Problem #3 The third problem: ten, partition it into two parts, and divide one by the other, and 4 is the result of the division. [fol. 36v]

Solution: by denoting one part of ten by x and the other by 10  x x

4 o 10  x

4 x o 5x

10  x

then

10

This is the third type of equation. By using the known algorithm for this case we obtain x 2 and 8 10  x 8 and 4. 2 Problem #4 The fourth problem: ten, partition it into two parts, and multiply the one part by itself and the other by nine, and they are equal. [fol. 36v]

Solution: by denoting one part of ten by x and the other by x 2 9(10  x) o x 2 90  9 x o x 2  9 x 90 .

10  x

then

This is the fourth type of equations. By using the known algorithm for this case we obtain x 6 and 10  x 4 . Verification: 62 9 ˜ 4 Problem #5 51 The fifth problem: ten, partition it into two parts, and multiply one of them by the other, resulting in 21. [fol. 36v]

Solution: by denoting one part of ten by x and the other by 10  x then x(10  x) 21 o 10 x  x 2 21 o x 2  21 10 x . This is the fifth type of equation. By using the known algorithm for this case, we obtain x 7 and 10  x 3 . Verification: 7 ˜ 3 21 .

51

This problem was presented by al-˜‚†Œ`& ee Mohamed ben Musa, The Algebra, pp. 6–13, 41.

76

THE EPISTLE OF THE NUMBER

Problem #6 The sixth problem: take an estate, add to it 8 Zuzim and multiply the sum by four, and the result is the multiplication of the estate by itself. [fol. 36v]

Solution: we obtain the bi-quadratic equation ( x 2  8) ˜ 4 x 2 ˜ x 2 52 We insert x instead of x 2 and obtain ( x  8)4 x ˜ x o 4 x  32 x 2 . This is the sixth type of equation. By using the known algorithm for this case we obtain x 8 and 10  x 2 . Verification: 16 ˜ 4 8 ˜ 8 Overcoming problematic expressions Isaac presents three rules to overcome the difficulty of dividing an inferior species by a higher one or dividing by a subtractive. In all the three following rules, the difficulties are overcome by transforming the problematic expression into a nonproblematic form. a Rule #1: b a b The author refers to problem #3 above, in which a-priori any choice of x as the divided term could yield a division of a lower species by a larger one. However, with the help of rule #1, one can easily observe how this difficulty is overcome. 10  x x

4 o 10  x

x 10  x

4ox

4 x o 5 x 10 o x

4(10  x) o x

o x 8, 10  x

2, 10  x

40  4 x o 5 x

8

40

2

Rule #2:

a b  b a

a2  b2 ab

Returning to problem #3: one inserts x for the one part and 10  x for the other part of ten. There is always one term which contains division by a subtractive and second in which a lower species is divided by a superior species. The proposed rule overcomes both difficulties:

52

There are no bi-quadratic equations in al-˜‚†Œ`&  Lévy, ‘L’algèbre Arabe’, p. 300. In al-˜‚|` \  'iyya justifies his calculations by geometric means; he does so for five problems, by using Euclid’s theorems. 179F

180F

63

Lit. ‘The book on mensuration and mensuration’. See Gad ben Ami Sarfatti, Mathematical Terminology in Hebrew Scientific Literature of the Middle Ages (in Hebrew, English summary), (Jerusalem 1968), pp. 68–69. 64 See Abraham Bar Hiyya, Chibbur ha-Meshicha veha-Tischboret, Lehrbuch der Geometrie des Abraham bar Chija, herausgegeben und mit Anmerkungen versehen von M. Guttmann, (Berlin 1912–1913), pp. 2–3.  /        +                     

0    * *   *    



   /  / 

0    /        3- - -*- 4           /   /        *   /        0    

 

/*  I have seen that the majority of the scholars of our generation in the land of ˆ are not familiar with the measurement of lands and they are not handy in their partition. They express a great disdain towards this matter and divide the lands between the heirs and associates by approximation and exaggeration. Since this is not [regarded] as being false and sinful, most of them are malicious and err shamefully and each one of them has sinned in the error in as much as the approximation is shameful. Do not [even] imagine that they [actually] measure and calculate, indeed they approximate and cheat. And I claim that they do not count in order, but they count by cheating, as is written: “I will make your oppressors eat their own flesh” (Isaiah 49:26), since there is no bigger fraud in the world perpetuated by them. It could occur that in their calculation the owner of one third gets one quarter and the owner of a quarter gets one third and you do not have greater theft and cheat than this.” 65

Lévy, ‘L’algèbre Arabe’, pp. 273–277.

90

THE EPISTLE OF THE NUMBER

The Book of Measures Around 1140, Abraham Ibn Ezra 66 composed The Book of Measures   , a text which includes problems on practical geometry. It derives from Arabic sources. It was translated into Latin during Ibn Ezra’s lifetime. The first part of the Hebrew text includes notes concerning arithmetical matters: the first ten numbers, the approximation of the square root and the usage of the root of 10 for the measurement of the circumference and area of circles. The next part presents a systematic study of geometry: points, lines, surfaces, volumes, various types of triangles, etc. It is in this part that Ibn Ezra uses the classical solution of quadratic equations. The application of algebraic procedures is present while algebraic vocabulary is absent, and again, geometric entities play the role of the unknowns. Most problems and their solutions are given in a general form and at times involve transformations. 67 Unlike Bar 'iyya’s text, Sefer ha-Middot does not include any geometric proof. 182F

The Book on Demonstration and Memorization of the Science of Problems Involving Dust Reckoning In May of 1271, Moses Ibn Tibbon, active between 1240 and 1283, translated into Hebrew Al-'a" twelfth century on arithmetic,

 -   ta‰ 

 Šil al- (The Book on Demonstration and Memorization of the Science of Problems Involving Dust Reckoning). To the best of our knowledge, three copies of this translation have survived. 68 For the very first time in a Hebrew treatise one encounters explicit algebraic vocabulary. This vocabulary consists of the following algebraic terminology:  /  / // , i.e. an estate, a thing, restoration, an unsolvable problem and an unknown. One first finds reference to algebra in the section on the addition of fractions in a passage dedicated to the addition of estates   + . There, one solves a linear equation, which is be solved by the rule of three and the rule of double false position. The term used for the restoration of the fraction, i.e. finding its inverse, is , but one of the scribes added by al-jabr in Hebrew letters to the margins. Further on, though, the term    is used for the restoration of fractions and not . It is interesting to note that the Arabic term al-jabr, which designates both the restoration of fraction as well as the algebraic restoration, was translated into two different terms in 66

Abraham Ibn Ezra is the author with all likelihood. Ibid, pp. 277–281. It is conjectured by Lévy and Burnett that the arithmetical part of Sefer ha-Middot may have been a first stage in the composition of Sefer ha-Mispar. See Tony Lévy and Charles Burnett, ‘Sefer ha-Middot: A Mid-Twelfth-Century Text on Arithmetic and Geometry Attributed to Abraham Ibn Ezra’, Aleph 6, (Jerusalem, 2006), 57–238. This text also contains at the end some rules for measuring distances with an astrolabe. 68 Oxford, Christ College Library, MS 189, fols. 1r–31v (IMHM 15581), Moscow, Russian State Library, MS Günzburg 30, fols. 121r–189v (IMHM 6711) and Rome, Biblioteca Apostolica Vaticana, MS Ebr. 396, fols. 1r–76v (IMHM 474). 67

A MATHEMATICAL COMMENTARY

91

Hebrew:  and  , respectively. 69 Algebra also appears in the part on sequences of natural numbers, even numbers, odd numbers, squares and cubes. There, the unknown is a sought member in the sequence of the natural numbers as well as the cubes. In the latter case, a bi-quadratic equation is formed, and it is solved by substitution of variables. The algebraic operations restoration and opposition are explicitly used in the process of solving quadratic equations. 70 In the text, the reader is guided to solve the problems by the first type of restoration        , probably referring to the fourth type of equations, i.e. the first among the composite ones. The very existence of this expression indicates that the general procedures of solution of the six canonical equations were either known to the reader, or explained in a different part of the text, which did not survive in Arabic or in Hebrew. Ibn al-Adab also explicitly mentions Al-'a’s name in Book I in a section concerning a shorter method for knowing the name of a number by its rank, a paragraph that we could identify in Ibn Tibbon’s Hebrew translation. 71 Isaac’s explicit mentioning of Al-'a name at this very point in The Epistle of the Number is crucial. Not only does it give us textual evidence for Isaac’s reading sources, but it also provides an interesting link between the Epistle to one of the few Hebrew texts which includes algebraic elements. We note that several algebraic terms are common to Al-'a Hebrew translation as well as The Epistle of the Number:  / /  , i.e. an estate, a thing and opposition, respectively. However, these Hebrew terms are not particular to Ibn Tibbon. Algebraic terms which are unique to Ibn Tibbon, such as     and  , i.e. an unsolvable problem and restoration, do not appear in the Epistle of the Number. This fact, combined with other Hebrew sources whose mathematical vocabulary was not adopted by Isaac, have led me to believe that Isaac only used Arabic sources for his composition of The Epistle of the Number, except for one possible reference to Sefer ha-Mispar. This is discussed chemin faisant in annotation of my translation. It is worth mentioning that the arithmetical contents of Al-'a book greatly overlap with those of Book I in the Epistle. This, however, comes as no surprise Al-'a treatise was probably a source of inspiration for the composition of Talkh A ml al-isb. 72 An inspection of the three copies of Ibn Tibbon’s translation reveals the following themes, which are explained by numerous examples: (i) the ranks of numbers and their names     (ii) the decimal place value system and its application on the ranks of numbers             (iii) the addition of numbers  +  (iv) subtraction of numbers from one another     (v) multiplication   (vi) ratio (vii) division of numbers   (viii) halving  (ix) doubling of numbers  (x) calculating 69 Isaac, on the other hand, translated al-jabr as  for both restoration of fractions as well as algebraic restoration. 70 Lévy, ‘L’algèbre Arabe’, pp. 281–286. 71 See fol. 2v in The Epistle. 72 Talk+, p. 33 (or p. 35 in the French part).

92

THE EPISTLE OF THE NUMBER

roots     . The Hebrew Epistle, mirroring the Talkh, has a different structure. For example, Al-'a explains each operation for all types of numbers (integers and fractions) in the same chapter, whereas The Epistle of the Number treats integers and fractions in separate chapters. Also, Al-'a text contains a much finer subdivision of the presented themes and it goes into greater detail than the Epistle, in particular in the chapter dedicated to fractions. For example, chapter 25 concerns the multiplication of the integer and two dissimilar fractions and the fraction of a fraction by an integer and two dissimilar fractions and a fraction of a fraction. This matter is not treated in the Hebrew Epistle. An anonymous commentary to T he Book of the Number Tony Lévy discovered a short mathematical text, whose date of composition is undetermined. It is an anonymous arithmetical text, 37 folios long, followed by an anonymous commentary to The Book of the Number  , whose short introduction (four folios long) includes algebraic elements. The text describes the algebraic species roots and squares and the two algebraic operations restoration and opposition. It presents the six canonical equations with the general procedure applied on six numerical examples. All the examples are identical to the ones found in al?}>"

 -Mukhtaar fi   -Jabr wa-al-  , except for one. The analysis of the mathematical terms in this Hebrew text and the comparison of those arithmetical elements to those within Book of Measures have led Tony Lévy to believe that this introduction might be connected to Abraham Ibn Ezra, at least it is contemporaneous with him. One can definitely define this passage within these four folios as being of algebraic nature, even though its presence is somewhat unexpected and unrelated to the mathematical elements which follow. 73 The main characteristics of each of the five treatises are described in the following table. 18F

73

For a detailed analysis, see Tony Lévy, ‘A Newly-Discovered Partial Hebrew Version of al-˜‚†Œ`& Algebra’, Aleph 2 (Jerusalem, 2002), 225–234.

A MATHEMATICAL COMMENTARY

Title of tract

Author/ translator

Place and time of composition The mathematical genre to which the text belongs Type of algebra used in the text (expressed in modern mathematical language) Number of problems in which algebraic algorithms are used The context in which algebra is used

Algebraic terms

T he Book on M esuration

The Book of M easures

     Abraham bar '||

 

93

The Book on Arithmetic by Al-

Anonymous text

(?) Abraham Ibn Ezra

Moses Ibn Tibbon (translated al'

?

Barcelona, first half of 12th century Practical geometry

Northern Italy(?), ca. 1140 Arithmetic and practical geometry

Montpellier, 1271

?

Arithmetic

Arithmetic

Arithmetic and algebra

Classical solution of quadratic equations

Classical solution of quadratic equations

Classical solution of quadratic equations

Solving the six canonical equations

6

18

7

6

Algebraic species, operations, Solution of quadratic equations, algebraic expressions Numerous

Plane Geometry (squares, rectangles, their diagonals and surface)

Plane Geometry (squares, rectangles, lozenges, their diagonals and surface) Non existent

Sequences of natural and cube numbers

?

Algebra per se as well as the solution of rhetorical problems

 A thing   Restoration  Restoration  

  An estate  A thing    Composite Cases   Squares  Roots     An unsolvable problem (referring to the fifth type of equation, which is not always solvable) Yes

Over fifty terms

Non-existent

Reduction  Opposed

Are general methods given?

No

Special traits

Geometric proof of the algebraic procedure follows five numerical examples

Yes

No, but there is reference to the first type of solution.

The Epistle of the Number   Isaac ben Solomon Ibn al  ated and annotated Ibn al- 

’s Talkh   - ) Syracuse, end of 14th century

Yes

CHAPTER 4: THE UNICUM The Epistle of the Number was composed at the end of the 14th century in Syracuse, Sicily, but the autograph is now lost. There is no indication to how many copies were made after the autograph and before our only surviving copy from mid-16th century Constantinople. The unicum is part of the codex Cambridge University Library MS Heb. Add. 492, which is described below.

CAMBRIDGE UNIVERSITY LIBRARY MS HEB. ADD. 492: CODICOLOGICAL AND PALAEOGRAPHICAL NOTES

The codex is of size 30cm x 21cm and it includes three paper tracts copied in the 16th century: Heb. 492.1: 38 folios long with foliation in Hindu-Arabic numerals. Each folio contains between 36 and 40 lines. Parts of the text had been affected by damp and corruption due to the oxidation of the brown ink, which resulted in holes in the place where letters had been written. 1 The handwriting is book hand semi-cursive Hispanic-Byzantine. Judging from the uniformity of the handwriting, it seems the text of The Epistle of the Number was copied by a scribe who emended the text in the margins, where we also find calculations and remarks by much later hands, some of which are in pencil. All 38 folios constitute separate entities, except for the first folio, which consists of a collage of two pages: on the recto side there is a folio with a mathematical problem concerning the measurement of distance between two birds flying from two towers. At the bottom of folio 1r we find the signature of Jacob Judah ‘Ali, probably the owner of (at least) the collated page (and the quire to which it had belonged). In folio 26v we find an ornamental monogram, which, albeit being difficult to read, reveals Judah’s name in it.

1

Stefan C. Reif, Hebrew Manuscripts, p. 318.

95

96

THE EPISTLE OF THE NUMBER

Cambridge University Library, MS Heb. Add. 492.1, fols. 1v–2r Reproduced by permission of the Syndics of Cambridge University Library

On the verso side of folio 1, we find the beginning of The Epistle of the Number at the end of a notebook. The text continues up to folio 35r, the end of a notebook. This is where the translation of Talkh A  -  ends. Folios 35v and 36r are blank. A new notebook which begins on folio 36v introduces a section which starts with “the commentator says”. This part includes a series of algebraic problems solved by algebraic methods presented in book II of the Epistle, and hence, this is probably an addendum to the main text, which may have been written later. This section is truncated two folios later, in folio 38v with the words “another example”.

THE UNICUM

97

Heb. Add. 492.2: 81 folios long with 32 lines each. It includes a treatise on arithmetic The Sought Numbers (  ) by the mathematician and exegete Elijah ben Abraham ha-Mizrai (Constantinople, c. 1450–1525). The contents of the text correspond to The Book of the Number ( ), by the same author, which was written in Constantinople in 1533. Heb. Add. 492.2 is divided into three sections, but it extends only to the beginning of the third section. Some of the materials complement the texts in 492.3. The owners of this manuscript are David Ibn Porena and Moses ben Isaac. Also, there are annotations and possibly diagrams made by Ha-Mizrai’s son, Israel. 2 190F

Heb. Add. 492.3 contains 59 folios with 31 lines each. It comprises four astronomical and arithmetical treatises, which are in part defective: x A Commentary on the Astrolabe (  *   ), a translation from the Arabic of Amad Ibn a-aˆ text. x The Sphere of the Celestial Sphere (   ), a translation from the Arabic of Qus I’ *" the celestial globe

 l-A bi lKura al-  (65 chapters). x Part of Elijah ben Abraham ha-Mizrai’s treatise on arithmetic The Number (). It is defective and carelessly numerated. x The third and final section of Ha-Mizrai’s treatise on arithmetic The Number (), in Responsa format. 3 19F

Errors in the unicum I have discerned two kinds of errors in The Epistle of the Number. The first one is of graphical-linguistic nature: instances of lapsus calami, including dittographies and haplographies, omission of necessary words or misrecognition of a word in the source copy. The second class of errors comprises calculation mistakes. A priori, it would be difficult to ascertain whether the numerical errors were made by Isaac himself or by scribes along the chain of 150 years of transmission of the text. However, given Isaac’s mathematical proficiency, which is evident from his transmission of Talkh   -  as well as his astronomical, exegetical and calendrical treatises, it would be more reasonable to assume that most (if not all) numerical errors emerged within during the transmission of the text.

2

Ibid, p. 318. A critical edition of the first part of the first article, with an examination of the mathematical contents in the larger context, the arithmetical terminology and the didactics have been published. See Segev, Stela, The Book of the Number by Elijah Mizrahi, a Textbook from the 15th Century, (in Hebrew), (Ph.D. dissertation, The Hebrew University of Jerusalem, Jerusalem, 2010). 3 Reif, Hebrew Manuscripts, p. 330.

98

THE EPISTLE OF THE NUMBER

Diagrams, tables and decorations in the unicum The non-verbal elements in our text include tables, diagrams, manicules 4 and decorations. The diagrams, such as of scales and multiplication tables for numbers and for algebraic operations visually demonstrate how to use the given algorithms. Manicules also have a didactic purpose. They call the reader’s attention to an important mathematical issue or the end of a section. They may even embed a humoristic facet. The following hand from folio 26v has six fingers!

Cambridge University Library, MS Heb. Add. 492.1, fol. 26v. Reproduced by permission of the Syndics of Cambridge University Library

4

I.e. pointing fingers.

CHAPTER 5: AN EDITION OF T HE E PISTLE OF THE

N UMBER NOTES TO THE EDITION The Hebrew edition aims to reflect an as correct as possible version of The Epistle of the Number, while attempting to approach the Urtext. Given the lacunae and numerous linguistic and arithmetical errors in the unicum, as well as dittographies and haplographies, I have decided to present the corrected version in the body of the edition, while the footnotes account for the lexical items which were removed or altered either by me or by one of the scribes involved in the original copying or revision of the text. Whenever I saw it fitting to omit a lexical item X, then it appears as (X) is the footnote. If X was erased from the text by a scribe, then it is presented in footnotes as (X). Words within square brackets [X] in the Hebrew text denote my own additions or corrections, which seem indispensable for the mathematical or linguistic correctness of the text. Starred words within square brackets [*] denote my reconstruction of illegible words in manuscript folios that sustained damage: erasure, holes, oxidation of ink or dampness stains. Words or letters appearing above the line or in the margins are inserted into triangular brackets . In the margins there are a few calculations in Hindu-Arabic numerals, not made by the scribe. They are accounted for in a footnote. Square parentheses containing a quest#„ ^ or a difficult interpretation and […] indicates lacunae in the Hebrew text. I have set the citations from Ibn al-’s Talkh A  -  in bold letters, but only the first time they appear. Hebrew vocalization is scarce in the unicum, but whenever it appears in the manuscript it is loyally preserved in the edition. In the manuscript there are very few punctuation marks, so to facilitate the understanding, I have added many others.

THE HEBREW IN THE EDITION The Epistle of the Number was written in medieval Hebrew. The beginning of the Epistle also includes a few biblical citations, but at times the verse is slightly altered to create a melia. Gender is frequently confused, as is often the case in medieval Hebrew, e.g. . There is a strong influence of the Arabic on the Hebrew syntactical forms, such as a conjugated verb in the imperative followed by another conjugated verb instead of an infinitive e.g.    (folio 17r) %& $ #$ '. At the same time we also find     (e.g. folio 11r). From Isaac’s introduction to The Epistle of the Number we learn that it is an isolated composition on the medieval Hebrew mathematical bookshelf, with no explicit connection to other translations from Arabic into Hebrew. Isaac indeed has 99

100

THE EPISTLE OF THE NUMBER

an independent style, and as I show along the annotations, he does not directly use Hebrew mathematical sources in his composition, only Arabic ones (only in folio 10r there is a possible allusion to Abraham Ibn Ezra’s Sefer ha-Mispar). This explains his novel Hebrew mathematical terminology not only in algebra, a possibly new field in the Jewish realm, but also in the well-established domain of arithmetic.

THE EDITION /       3143I4        , *               1     0  *          0      0          

  0      / - 

 /    0      2 .    /                        

   0  3          0       

 0          *  0   /  *   /          0         /   - -    0     /       4 , 

+      /   0             -   6     5              /  "/   0      /*               

0  '                   7 ,   0    

            0   /    0  3* 4   0        0           1

Joel 2:20. Exodus 34:34. 3 Exodus 34:33. 4 Psalms 120:5. 5 Psalms 42:8. 6 In Psalms 44:26 we find the semantically equivalent    “our soul is bowed down to the dust…” Isaac seems to have slightly adaped the verse, according to the rules known as melia. 7 Job 21:8. 2

THE HEBREW EDITION

101

        

             /         "   0         0           /        *   0

                    

   /             0              .    * 

 0        0  

          

   *    *      /      - .      - 

 0         -

 0            

         0                      

 0       /     

                                0         

            0      /     

           +  /      /-  0                                +                      0    0 / 

  0 ' -  0         /   0  /   +   - /     ' - .      0 .

     

    0          0-    /   "/-   -

0         /   -               3 4      0  / 0  / 

 

 

 

/- /   /        "  

       

0           

 -' - ' - 

/  /-   -   //   -  "     -'  "

     

0 /  0     -    /       

102

THE EPISTLE OF THE NUMBER

       3 24                 0      8 3 

 

4

   '     

    - '  - '  -  -